Theorem vtoclf 1377

Description: Implicit substitution of a class for a set variable. This is a generalization of chv2 850.

Hypotheses

Ref	Expression
vtoclf.1	⊢ (ψ → ∀xψ)
vtoclf.2	⊢ A ∈ V
vtoclf.3	⊢ (x = A → (φ ↔ ψ))
vtoclf.4	⊢ φ

Assertion

Ref	Expression
vtoclf	⊢ ψ

Distinct variable group(s):   x,A

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. . 3 ⊢ (ψ → ∀xψ)
2		vtoclf.2	. . . . 5 ⊢ A ∈ V
3	2	isseti 1352	. . . 4 ⊢ ∃x x = A
4		vtoclf.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
5	4	biimpd 135	. . . . 5 ⊢ (x = A → (φ → ψ))
6	5	19.22i 723	. . . 4 ⊢ (∃x x = A → ∃x(φ → ψ))
7	3, 6	ax-mp 6	. . 3 ⊢ ∃x(φ → ψ)
8	1, 7	19.36i 758	. 2 ⊢ (∀xφ → ψ)
9		vtoclf.4	. 2 ⊢ φ
10	8, 9	mpg 684	1 ⊢ ψ

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  vtocl 1378  axrep2 1474  zfcndrep 3760

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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