Theorem bj-axseprep 38032

Description: Axiom of separation (universal closure of ax-sep 5248) from a weak form of the axiom of replacement requiring that the functional relation in it be a (total) function and the weak emptyset axiom (existence of an empty set provided existence of a set), as written in the theorem's hypotheses.

This result shows that the weak emptyset axiom is not only the result of a cheap way to avoid an axiom redundancy (in this case, the existence axiom extru 2007) by adding it as an antecedent, but also permits to prove nontrivial results that hold in nonnecessarily nonempty universes.

This proof is by cases so is not intuitionistic. The statement does not require a nonempty universe; most of the proof does not either, and the parts that do (e.g., near sb8ef 2384 and sbequ12r 2287 and eueq2 3667) could be reworked to avoid it. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1959. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-axseprep.axnulw	⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
bj-axseprep.axrep	⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓))
bj-axseprep.ps	⊢ (𝜓 ↔ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))

Assertion

Ref	Expression
bj-axseprep	⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))

Distinct variable groups:   𝑡,𝑎,𝑥,𝑦,𝑧   𝜑,𝑎,𝑡,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑥, 𝑦, 𝑧, 𝑡, 𝑎)

Proof of Theorem bj-axseprep

Step	Hyp	Ref	Expression
1		ax5e 1944	. . . 4 ⊢ (∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
2	1	ax-gen 1827	. . 3 ⊢ ∀𝑥(∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
3		bj-eximcom 37558	. . . . 5 ⊢ (∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → (∀𝑎∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
4		bj-axseprep.axrep	. . . . . . . . 9 ⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓))
5		bj-axseprep.ps	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
6	5	eubii 2610	. . . . . . . . . . . 12 ⊢ (∃!𝑡𝜓 ↔ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
7	6	ralbii 3108	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
8	5	rexbii 3109	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝑥 𝜓 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
9	8	bibi2i 340	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ (𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
10	9	albii 1851	. . . . . . . . . . . 12 ⊢ (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ ∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
11	10	exbii 1880	. . . . . . . . . . 11 ⊢ (∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
12	7, 11	imbi12i 353	. . . . . . . . . 10 ⊢ ((∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓)) ↔ (∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))))
13	12	albii 1851	. . . . . . . . 9 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓)) ↔ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))))
14	4, 13	mpbi 233	. . . . . . . 8 ⊢ ∀𝑥(∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
15		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		vex 3454	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
17	15, 16	eueq2 3667	. . . . . . . . . 10 ⊢ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
18	17	rgenw 3080	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
19	18	ax-gen 1827	. . . . . . . 8 ⊢ ∀𝑥∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
20	14, 19	bj-almp 37523	. . . . . . 7 ⊢ ∀𝑥∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
21	20	ax-gen 1827	. . . . . 6 ⊢ ∀𝑎∀𝑥∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
22		alcom 2195	. . . . . 6 ⊢ (∀𝑎∀𝑥∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ∀𝑥∀𝑎∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
23	21, 22	mpbi 233	. . . . 5 ⊢ ∀𝑥∀𝑎∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
24	3, 23	bj-almpig 37532	. . . 4 ⊢ ∀𝑥(∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → ∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
25		df-rex 3087	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝜑 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝜑))
26		nfv 1946	. . . . . . . 8 ⊢ Ⅎ𝑎(𝑧 ∈ 𝑥 ∧ 𝜑)
27	26	sb8ef 2384	. . . . . . 7 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑎[𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
28	25, 27	bitri 278	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝜑 ↔ ∃𝑎[𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
29		df-rex 3087	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
30		andi 1025	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ((𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ (𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
31	30	exbii 1880	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ∃𝑧((𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ (𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
32		19.43 1914	. . . . . . . . . . . . . 14 ⊢ (∃𝑧((𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ (𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
33	29, 31, 32	3bitri 300	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
34		equcom 2050	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑡 ↔ 𝑡 = 𝑧)
35	34	anbi1i 636	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑡 = 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
36		ancom 466	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 = 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ ((𝑧 ∈ 𝑥 ∧ 𝜑) ∧ 𝑡 = 𝑧))
37		anass 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ 𝑥 ∧ 𝜑) ∧ 𝑡 = 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
38	35, 36, 37	3bitri 300	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
39	38	exbii 1880	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
40	39	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
41	40	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
42		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → 𝑡 = 𝑎)
43	42	exlimiv 1962	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → 𝑡 = 𝑎)
44		sbequ 2119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑡 → ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) ↔ [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
45	44	biimpd 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑡 → ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
46	45	equcoms 2052	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑎 → ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
47	46	com12 33	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (𝑡 = 𝑎 → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
48		sb5 2309	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
49	47, 48	imbitrdi 254	. . . . . . . . . . . . . . . 16 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (𝑡 = 𝑎 → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
50	43, 49	syl5 35	. . . . . . . . . . . . . . 15 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
51	41, 50	jaod 873	. . . . . . . . . . . . . 14 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
52		orc 881	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
53	39, 52	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
54	51, 53	impbid1 228	. . . . . . . . . . . . 13 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
55	33, 54	bitrid 286	. . . . . . . . . . . 12 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
56	55	bibi2d 345	. . . . . . . . . . 11 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ (𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))))
57	56	biimpd 232	. . . . . . . . . 10 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → (𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))))
58	57	alimdv 1948	. . . . . . . . 9 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))))
59		nfv 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑡 ∈ 𝑦
60		nfe1 2186	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))
61	59, 60	nfbi 1935	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
62		nfv 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))
63		elequ1 2151	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
64	48	bicomi 227	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
65		sbequ12r 2287	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → ([𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
66	64, 65	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
67	63, 66	bibi12d 348	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → ((𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))) ↔ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
68	61, 62, 67	cbvalv1 2370	. . . . . . . . 9 ⊢ (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
69	58, 68	imbitrdi 254	. . . . . . . 8 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
70	69	eximdv 1949	. . . . . . 7 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
71	70	eximi 1867	. . . . . 6 ⊢ (∃𝑎[𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
72	28, 71	sylbi 220	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
73	72	ax-gen 1827	. . . 4 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
74	24, 73	barbara 2687	. . 3 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
75	2, 74	barbara 2687	. 2 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
76		ralnex 3088	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ¬ ∃𝑧 ∈ 𝑥 𝜑)
77		df-ral 3077	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑))
78		df-ral 3077	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ⊥ ↔ ∀𝑧(𝑧 ∈ 𝑦 → ⊥))
79		dfnot 1588	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 → ⊥))
80	79	bicomi 227	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 → ⊥) ↔ ¬ 𝑧 ∈ 𝑦)
81		imnan 405	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 → ¬ 𝜑) ↔ ¬ (𝑧 ∈ 𝑥 ∧ 𝜑))
82		pm5.21 837	. . . . . . . . . 10 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ¬ (𝑧 ∈ 𝑥 ∧ 𝜑)) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
83	80, 81, 82	syl2anb 610	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 → ⊥) ∧ (𝑧 ∈ 𝑥 → ¬ 𝜑)) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
84	83	expcom 419	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑥 → ¬ 𝜑) → ((𝑧 ∈ 𝑦 → ⊥) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
85	84	al2imi 1847	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑) → (∀𝑧(𝑧 ∈ 𝑦 → ⊥) → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
86	78, 85	biimtrid 245	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑) → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
87	77, 86	sylbi 220	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
88	76, 87	sylbir 238	. . . 4 ⊢ (¬ ∃𝑧 ∈ 𝑥 𝜑 → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
89	88	eximdv 1949	. . 3 ⊢ (¬ ∃𝑧 ∈ 𝑥 𝜑 → (∃𝑦∀𝑧 ∈ 𝑦 ⊥ → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
90		bj-axseprep.axnulw	. . . 4 ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
91		bj-alextruim 37578	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥))
92	90, 91	mpbir 234	. . 3 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
93	89, 92	bj-almpig 37532	. 2 ⊢ ∀𝑥(¬ ∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
94		pm2.61 194	. . 3 ⊢ ((∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → ((¬ ∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
95	94	al2imi 1847	. 2 ⊢ (∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → (∀𝑥(¬ ∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))) → ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
96	75, 93, 95	mp2 9	1 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861  ∀wal 1567   = wceq 1569  ⊤wtru 1570  ⊥wfal 1581  ∃wex 1811  [wsb 2098   ∈ wcel 2144  ∃!weu 2593  ∀wral 3076  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452

This theorem is used by: (None)