Theorem spcgv 2940

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Hypothesis

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

Assertion

Ref	Expression
spcgv	⊢ (A ∈ V → (∀xφ → ψ))

Distinct variable groups:   ψ,x   x,A

Allowed substitution hints:   φ(x)   V(x)

Proof of Theorem spcgv

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		spcgv.1	. 2 ⊢ (x = A → (φ ↔ ψ))
4	1, 2, 3	spcgf 2935	1 ⊢ (A ∈ V → (∀xφ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = 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-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-v 2862

This theorem is referenced by:  spcv  2946  mob2  3017  intss1  3942  dfiin2g  4001  sfintfin  4533  funmo  5126  frd  5923