Theorem axrep3 5209

Description: Axiom of Replacement slightly strengthened from axrep2 5208; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2372. (Revised by BJ, 31-May-2019.)

Assertion

Ref	Expression
axrep3	⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

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

Proof of Theorem axrep3

Step	Hyp	Ref	Expression
1		nfe1 2149	. . . 4 ⊢ Ⅎ𝑦∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦)
2		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑥
3		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑤
4		nfa1 2150	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦𝜑
5	3, 4	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
6	5	nfex 2322	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
7	2, 6	nfbi 1907	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
8	7	nfal 2321	. . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
9	1, 8	nfim 1900	. . 3 ⊢ Ⅎ𝑦(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
10	9	nfex 2322	. 2 ⊢ Ⅎ𝑦∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
11		axreplem 5207	. 2 ⊢ (𝑦 = 𝑤 → (∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) ↔ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))))
12		axrep2 5208	. 2 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
13	10, 11, 12	chvarfv 2236	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-rep 5205

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by:  axrep4  5210