Theorem vtocld 2782

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
vtocld.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vtocld.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
vtocld.3	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
vtocld	⊢ (𝜑 → 𝜒)

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

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

Proof of Theorem vtocld

Step	Hyp	Ref	Expression
1		vtocld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		vtocld.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3		vtocld.3	. 2 ⊢ (𝜑 → 𝜓)
4		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜑
5		nfcvd 2313	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfvd 1522	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
7	1, 2, 3, 4, 5, 6	vtocldf 2781	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  funfvima3  5726  isbth  6940  frec2uzuzd  10345