Theorem vtoclf 2909

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1986. (Contributed by NM, 30-Aug-1993.)

Hypotheses

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

Assertion

Ref	Expression
vtoclf	⊢ ψ

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem vtoclf

Step	Hyp	Ref	Expression
1		vtoclf.1	. . 3 ⊢ Ⅎxψ
2		vtoclf.2	. . . . 5 ⊢ A ∈ V
3	2	isseti 2866	. . . 4 ⊢ ∃x x = A
4		vtoclf.3	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
5	4	biimpd 198	. . . . 5 ⊢ (x = A → (φ → ψ))
6	5	eximi 1576	. . . 4 ⊢ (∃x x = A → ∃x(φ → ψ))
7	3, 6	ax-mp 5	. . 3 ⊢ ∃x(φ → ψ)
8	1, 7	19.36i 1872	. 2 ⊢ (∀xφ → ψ)
9		vtoclf.4	. 2 ⊢ φ
10	8, 9	mpg 1548	1 ⊢ ψ

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is used by:  vtocl  2910