Theorem vtocld 2907

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 1619	. 2 ⊢ Ⅎxφ
5		nfcvd 2490	. 2 ⊢ (φ → ℲxA)
6		nfvd 1620	. 2 ⊢ (φ → Ⅎxχ)
7	1, 2, 3, 4, 5, 6	vtocldf 2906	1 ⊢ (φ → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by: (None)