Theorem axrep3

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

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

Proof

Step	Hyp	Ref	Expression
1		nfe1	⊢ Ⅎ 𝑦 ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 )
2		nfv	⊢ Ⅎ 𝑦 𝑧 ∈ 𝑥
3		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ 𝑤
4		nfa1	⊢ Ⅎ 𝑦 ∀ 𝑦 𝜑
5	3 4	nfan	⊢ Ⅎ 𝑦 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 )
6	5	nfex	⊢ Ⅎ 𝑦 ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 )
7	2 6	nfbi	⊢ Ⅎ 𝑦 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) )
8	7	nfal	⊢ Ⅎ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) )
9	1 8	nfim	⊢ Ⅎ 𝑦 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) ) )
10	9	nfex	⊢ Ⅎ 𝑦 ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) ) )
11		axreplem	⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) ) ↔ ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) ) ) ) )
12		axrep2	⊢ ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑦 ∧ ∀ 𝑦 𝜑 ) ) )
13	10 11 12	chvarfv	⊢ ∃ 𝑥 ( ∃ 𝑦 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑦 ) → ∀ 𝑧 ( 𝑧 ∈ 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑤 ∧ ∀ 𝑦 𝜑 ) ) )