Theorem spcimgft 2931

Description: A closed version of spcimgf 2933. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

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

Assertion

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

Proof of Theorem spcimgft

Step	Hyp	Ref	Expression
1		elex 2868	. 2 ⊢ (A ∈ B → A ∈ V)
2		spcimgft.2	. . . . 5 ⊢ ℲxA
3	2	issetf 2865	. . . 4 ⊢ (A ∈ V ↔ ∃x x = A)
4		exim 1575	. . . 4 ⊢ (∀x(x = A → (φ → ψ)) → (∃x x = A → ∃x(φ → ψ)))
5	3, 4	syl5bi 208	. . 3 ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ V → ∃x(φ → ψ)))
6		spcimgft.1	. . . 4 ⊢ Ⅎxψ
7	6	19.36 1871	. . 3 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))
8	5, 7	syl6ib 217	. 2 ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ V → (∀xφ → ψ)))
9	1, 8	syl5 28	1 ⊢ (∀x(x = A → (φ → ψ)) → (A ∈ B → (∀xφ → ψ)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860

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:  spcgft  2932  spcimgf  2933  spcimdv  2937