Theorem spcgf 3538

Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Hypotheses

Ref	Expression
spcgf.1	⊢ Ⅎ𝑥𝐴
spcgf.2	⊢ Ⅎ𝑥𝜓
spcgf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem spcgf

Step	Hyp	Ref	Expression
1		spcgf.2	. . 3 ⊢ Ⅎ𝑥𝜓
2		spcgf.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	spcgft 3535	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpg 1799	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443

This theorem is referenced by:  spcegf  3539  spcgvOLD  3544  rspc  3559  elabgt  3609  eusvnf  5258  gropd  26824  grstructd  26825