Theorem vtocld 3556

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 1911	. 2 ⊢ Ⅎ𝑥𝜑
5		nfcvd 2978	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐴)
6		nfvd 1912	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
7	1, 2, 3, 4, 5, 6	vtocldf 3555	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by:  vtocl2d  3557  lmatfval  31079  lmatcl  31081  dvgrat  40642  dfatbrafv2b  43443  fnbrafv2b  43446