Theorem vtoclb 2794

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 235	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃)))
5		vtoclb.4	. 2 ⊢ (𝜑 ↔ 𝜓)
6	1, 4, 5	vtocl 2791	1 ⊢ (𝜒 ↔ 𝜃)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739

This theorem is referenced by:  alexeq  2863  sbss  3531