Theorem spcgft 2932

Description: A closed version of spcgf 2935. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimgft.1	⊢ Ⅎxψ
spcimgft.2	⊢ ℲxA

Assertion

Ref	Expression
spcgft	⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀xφ → ψ)))

Proof of Theorem spcgft

Step	Hyp	Ref	Expression
1		bi1 178	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	imim2i 13	. . 3 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (φ → ψ)))
3	2	alimi 1559	. 2 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(x = A → (φ → ψ)))
4		spcimgft.1	. . 3 ⊢ Ⅎxψ
5		spcimgft.2	. . 3 ⊢ ℲxA
6	4, 5	spcimgft 2931	. 2 ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ B → (∀xφ → ψ)))
7	3, 6	syl 15	1 ⊢ (∀x(x = A → (φ ↔ ψ)) → (A ∈ B → (∀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:  spcgf  2935  rspct  2949