Theorem vtocld 2791

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 1528	. 2 ⊢ Ⅎ𝑥𝜑
5		nfcvd 2320	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfvd 1529	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
7	1, 2, 3, 4, 5, 6	vtocldf 2790	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148

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-7 1448  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-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by:  funfvima3  5752  isbth  6968  frec2uzuzd  10404  setscomd  12505