Theorem spcgf 3581

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 3578	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
4		spcgf.3	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpg 1798	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  Ⅎwnfc 2882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475

This theorem is referenced by:  spcegf  3582  rspc  3600  elabgtOLD  3663  eusvnf  5390  gropd  28724  grstructd  28725