Theorem spcgf 2935

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 2932	. 2 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ V → (∀xφ → ψ)))
4		spcgf.3	. 2 ⊢ (x = A → (φ ↔ ψ))
5	3, 4	mpg 1548	1 ⊢ (A ∈ V → (∀xφ → ψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This proof depends on 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 proof 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 used by:  spcegf  2936  spcgv  2940  rspc  2950  elabgt  2983