Theorem spcgf 2889

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 2884	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpg 1500	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398  Ⅎwnf 1509   ∈ wcel 2202  Ⅎwnfc 2362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  spcgv  2894  rspc  2905  elabgt  2948  eusvnf  4556  mpofvex  6379  modom  7037  gropd  15971  grstructd2dom  15972