Theorem bj-axseprep 37313

Description: Axiom of separation (universal closure of ax-sep 5243) 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 1977) 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 2360 and sbequ12r 2260 and eueq2 3670) could be reworked to avoid it. Proof modifications should not introduce steps relying on a nonempty universe, like alrimiv 1929. (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 1914	. . . 4 ⊢ (∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
2	1	ax-gen 1797	. . 3 ⊢ ∀𝑥(∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
3		bj-eximcom 36870	. . . . 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 2586	. . . . . . . . . . . 12 ⊢ (∃!𝑡𝜓 ↔ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
7	6	ralbii 3084	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
8	5	rexbii 3085	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝑥 𝜓 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
9	8	bibi2i 337	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ (𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
10	9	albii 1821	. . . . . . . . . . . 12 ⊢ (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ ∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
11	10	exbii 1850	. . . . . . . . . . 11 ⊢ (∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 𝜓) ↔ ∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
12	7, 11	imbi12i 350	. . . . . . . . . 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 1821	. . . . . . . . 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 230	. . . . . . . 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 3446	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		vex 3446	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
17	15, 16	eueq2 3670	. . . . . . . . . 10 ⊢ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
18	17	rgenw 3056	. . . . . . . . 9 ⊢ ∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
19	18	ax-gen 1797	. . . . . . . 8 ⊢ ∀𝑥∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
20	14, 19	bj-almp 36832	. . . . . . 7 ⊢ ∀𝑥∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
21	20	ax-gen 1797	. . . . . 6 ⊢ ∀𝑎∀𝑥∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
22		alcom 2165	. . . . . 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 230	. . . . 5 ⊢ ∀𝑥∀𝑎∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
24	3, 23	bj-almpig 36835	. . . 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 3063	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝜑 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝜑))
26		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑎(𝑧 ∈ 𝑥 ∧ 𝜑)
27	26	sb8ef 2360	. . . . . . 7 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑎[𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
28	25, 27	bitri 275	. . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝜑 ↔ ∃𝑎[𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
29		df-rex 3063	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
30		andi 1010	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ((𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ (𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
31	30	exbii 1850	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ∃𝑧((𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ (𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
32		19.43 1884	. . . . . . . . . . . . . 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 297	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
34		equcom 2020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑡 ↔ 𝑡 = 𝑧)
35	34	anbi1i 625	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑡 = 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
36		ancom 460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑡 = 𝑧 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ ((𝑧 ∈ 𝑥 ∧ 𝜑) ∧ 𝑡 = 𝑧))
37		anass 468	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ 𝑥 ∧ 𝜑) ∧ 𝑡 = 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
38	35, 36, 37	3bitri 297	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
39	38	exbii 1850	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)))
40	39	biimpri 228	. . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → 𝑡 = 𝑎)
43	42	exlimiv 1932	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → 𝑡 = 𝑎)
44		sbequi 2090	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑡 → ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
45	44	equcoms 2022	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑎 → ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
46	45	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (𝑡 = 𝑎 → [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑)))
47		sb5 2283	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
48	46, 47	imbitrdi 251	. . . . . . . . . . . . . . . 16 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (𝑡 = 𝑎 → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
49	43, 48	syl5 34	. . . . . . . . . . . . . . 15 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎)) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
50	41, 49	jaod 860	. . . . . . . . . . . . . 14 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
51		orc 868	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
52	39, 51	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) → (∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))))
53	50, 52	impbid1 225	. . . . . . . . . . . . 13 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((∃𝑧(𝑧 ∈ 𝑥 ∧ (𝜑 ∧ 𝑡 = 𝑧)) ∨ ∃𝑧(𝑧 ∈ 𝑥 ∧ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
54	33, 53	bitrid 283	. . . . . . . . . . . 12 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)) ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))))
55	54	bibi2d 342	. . . . . . . . . . 11 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → ((𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) ↔ (𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))))
56	55	biimpd 229	. . . . . . . . . 10 ⊢ ([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	alimdv 1918	. . . . . . . . 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)))))
58		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑡 ∈ 𝑦
59		nfe1 2156	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))
60	58, 59	nfbi 1905	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)))
61		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))
62		elequ1 2121	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
63	47	bicomi 224	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ [𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑))
64		sbequ12r 2260	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → ([𝑡 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
65	63, 64	bitrid 283	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑)) ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
66	62, 65	bibi12d 345	. . . . . . . . . 10 ⊢ (𝑡 = 𝑧 → ((𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))) ↔ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
67	60, 61, 66	cbvalv1 2346	. . . . . . . . 9 ⊢ (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑡 ∧ (𝑧 ∈ 𝑥 ∧ 𝜑))) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
68	57, 67	imbitrdi 251	. . . . . . . 8 ⊢ ([𝑎 / 𝑧](𝑧 ∈ 𝑥 ∧ 𝜑) → (∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
69	68	eximdv 1919	. . . . . . 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))))
70	69	eximi 1837	. . . . . 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))))
71	28, 70	sylbi 217	. . . . 5 ⊢ (∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎(∃𝑦∀𝑡(𝑡 ∈ 𝑦 ↔ ∃𝑧 ∈ 𝑥 ((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
72	71	ax-gen 1797	. . . 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))))
73	24, 72	barbara 2664	. . 3 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
74	2, 73	barbara 2664	. 2 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
75		ralnex 3064	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ¬ ∃𝑧 ∈ 𝑥 𝜑)
76		df-ral 3053	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑))
77		df-ral 3053	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ⊥ ↔ ∀𝑧(𝑧 ∈ 𝑦 → ⊥))
78		dfnot 1561	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 → ⊥))
79	78	bicomi 224	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 → ⊥) ↔ ¬ 𝑧 ∈ 𝑦)
80		imnan 399	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑥 → ¬ 𝜑) ↔ ¬ (𝑧 ∈ 𝑥 ∧ 𝜑))
81		pm5.21 825	. . . . . . . . . 10 ⊢ ((¬ 𝑧 ∈ 𝑦 ∧ ¬ (𝑧 ∈ 𝑥 ∧ 𝜑)) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
82	79, 80, 81	syl2anb 599	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 → ⊥) ∧ (𝑧 ∈ 𝑥 → ¬ 𝜑)) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
83	82	expcom 413	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑥 → ¬ 𝜑) → ((𝑧 ∈ 𝑦 → ⊥) → (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
84	83	al2imi 1817	. . . . . . 7 ⊢ (∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑) → (∀𝑧(𝑧 ∈ 𝑦 → ⊥) → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
85	77, 84	biimtrid 242	. . . . . 6 ⊢ (∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑) → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
86	76, 85	sylbi 217	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
87	75, 86	sylbir 235	. . . 4 ⊢ (¬ ∃𝑧 ∈ 𝑥 𝜑 → (∀𝑧 ∈ 𝑦 ⊥ → ∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
88	87	eximdv 1919	. . 3 ⊢ (¬ ∃𝑧 ∈ 𝑥 𝜑 → (∃𝑦∀𝑧 ∈ 𝑦 ⊥ → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))))
89		bj-axseprep.axnulw	. . . 4 ⊢ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥)
90		bj-alextruim 36873	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥))
91	89, 90	mpbir 231	. . 3 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
92	88, 91	bj-almpig 36835	. 2 ⊢ ∀𝑥(¬ ∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
93		pm2.61 192	. . 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))))
94	93	al2imi 1817	. 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))))
95	74, 92, 94	mp2 9	1 ⊢ ∀𝑥∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542  ⊤wtru 1543  ⊥wfal 1554  ∃wex 1781  [wsb 2068   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444

This theorem is referenced by: (None)