Theorem rspccva 2955

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
rspcv.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
rspccva	⊢ ((∀x ∈ B φ ∧ A ∈ B) → ψ)

Distinct variable groups:   x,A   x,B   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem rspccva

Step	Hyp	Ref	Expression
1		rspcv.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
2	1	rspcv 2952	. 2 ⊢ (A ∈ B → (∀x ∈ B φ → ψ))
3	2	impcom 419	1 ⊢ ((∀x ∈ B φ ∧ A ∈ B) → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862

This theorem is referenced by:  disjne  3597  foelrn  5426  fconstfv  5457  nchoicelem17  6306