Theorem vtoclb 2913

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.)

Hypotheses

Ref	Expression
vtoclb.1	⊢ A ∈ V
vtoclb.2	⊢ (x = A → (φ ↔ χ))
vtoclb.3	⊢ (x = A → (ψ ↔ θ))
vtoclb.4	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
vtoclb	⊢ (χ ↔ θ)

Distinct variable groups:   x,A   χ,x   θ,x

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

Proof of Theorem vtoclb

Step	Hyp	Ref	Expression
1		vtoclb.1	. 2 ⊢ A ∈ V
2		vtoclb.2	. . 3 ⊢ (x = A → (φ ↔ χ))
3		vtoclb.3	. . 3 ⊢ (x = A → (ψ ↔ θ))
4	2, 3	bibi12d 312	. 2 ⊢ (x = A → ((φ ↔ ψ) ↔ (χ ↔ θ)))
5		vtoclb.4	. 2 ⊢ (φ ↔ ψ)
6	1, 4, 5	vtocl 2910	1 ⊢ (χ ↔ θ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This theorem was proved from 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 theorem 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 referenced by:  alexeq  2969  sbss  3660