Theorem bj-axseprep 37369

Description: Axiom of separation (universal closure of ax-sep 5220) 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 2358 and sbequ12r 2259 and eueq2 3653) 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 36899	. . . . 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 2584	. . . . . . . . . . . 12 ⊢ (∃!𝑡𝜓 ↔ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
7	6	ralbii 3081	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑡𝜓 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎)))
8	5	rexbii 3082	. . . . . . . . . . . . . 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 3431	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		vex 3431	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
17	15, 16	eueq2 3653	. . . . . . . . . 10 ⊢ ∃!𝑡((𝜑 ∧ 𝑡 = 𝑧) ∨ (¬ 𝜑 ∧ 𝑡 = 𝑎))
18	17	rgenw 3053	. . . . . . . . 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 36864	. . . . . . 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 36873	. . . 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 3060	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝑥 𝜑 ↔ ∃𝑧(𝑧 ∈ 𝑥 ∧ 𝜑))
26		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑎(𝑧 ∈ 𝑥 ∧ 𝜑)
27	26	sb8ef 2358	. . . . . . 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 3060	. . . . . . . . . . . . . 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 2282	. . . . . . . . . . . . . . . . 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 2259	. . . . . . . . . . . 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 2344	. . . . . . . . 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 2662	. . 3 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑎∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
74	2, 73	barbara 2662	. 2 ⊢ ∀𝑥(∃𝑧 ∈ 𝑥 𝜑 → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ 𝜑)))
75		ralnex 3061	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ¬ ∃𝑧 ∈ 𝑥 𝜑)
76		df-ral 3050	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝜑 ↔ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝜑))
77		df-ral 3050	. . . . . . 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 36919	. . . 4 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥ ↔ (∃𝑥⊤ → ∃𝑦∀𝑧 ∈ 𝑦 ⊥))
91	89, 90	mpbir 231	. . 3 ⊢ ∀𝑥∃𝑦∀𝑧 ∈ 𝑦 ⊥
92	88, 91	bj-almpig 36873	. 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 2567  ∀wral 3049  ∃wrex 3059

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 2184  ax-ext 2707

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3050  df-rex 3060  df-v 3429

This theorem is referenced by: (None)