Theorem axrep3 5228

Description: Axiom of Replacement slightly strengthened from axrep2 5227; 𝑤 may occur free in 𝜑. (Contributed by NM, 2-Jan-1997.) Remove dependency on ax-13 2402. (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 2183	. . . 4 ⊢ Ⅎ𝑦∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦)
2		nfv 1933	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝑥
3		nfv 1933	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝑤
4		nfa1 2184	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦𝜑
5	3, 4	nfan 1918	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
6	5	nfex 2355	. . . . . 6 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)
7	2, 6	nfbi 1922	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
8	7	nfal 2354	. . . 4 ⊢ Ⅎ𝑦∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑))
9	1, 8	nfim 1915	. . 3 ⊢ Ⅎ𝑦(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
10	9	nfex 2355	. 2 ⊢ Ⅎ𝑦∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))
11		axreplem 5226	. 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 5227	. 2 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))
13	10, 11, 12	chvarfv 2274	1 ⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-rep 5224

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803

This theorem is referenced by:  axrep4OLD  5231