Theorem axrep4 5159

Description: A more traditional version of the Axiom of Replacement. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
axrep4.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
axrep4	⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem axrep4

Step	Hyp	Ref	Expression
1		axrep3 5158	. . 3 ⊢ ∃𝑥(∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
2	1	19.35i 1879	. 2 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)))
3		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝑥
4		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤
5		nfa1 2152	. . . . . . 7 ⊢ Ⅎ𝑧∀𝑧𝜑
6	4, 5	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)
7	6	nfex 2332	. . . . 5 ⊢ Ⅎ𝑧∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)
8	3, 7	nfbi 1904	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑))
9	8	nfal 2331	. . 3 ⊢ Ⅎ𝑧∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑))
10		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧
11		nfe1 2151	. . . . 5 ⊢ Ⅎ𝑥∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)
12	10, 11	nfbi 1904	. . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
13	12	nfal 2331	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
14		elequ2 2126	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
15		axrep4.1	. . . . . . . . 9 ⊢ Ⅎ𝑧𝜑
16	15	19.3 2200	. . . . . . . 8 ⊢ (∀𝑧𝜑 ↔ 𝜑)
17	16	anbi2i 625	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ (𝑥 ∈ 𝑤 ∧ 𝜑))
18	17	exbii 1849	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))
19	18	a1i 11	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑) ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
20	14, 19	bibi12d 349	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
21	20	albidv 1921	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑))))
22	9, 13, 21	cbvexv1 2351	. 2 ⊢ (∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑧𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))
23	2, 22	sylib 221	1 ⊢ (∀𝑥∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785

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 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-rep 5154

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786

This theorem is referenced by:  axrep5  5160  axprlem3  5291  funimaexg  6410