Theorem vtocld 2875

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 1581	. 2 ⊢ Ⅎ𝑥𝜑
5		nfcvd 2393	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfvd 1582	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
7	1, 2, 3, 4, 5, 6	vtocldf 2874	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  funfvima3  5942  isbth  7274  frec2uzuzd  10817  setscomd  13371