Theorem spcimgfi1OLD 3519

Description: Obsolete version of spcimgfi1 3518 as of 27-Jul-2025. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spcimgfi1.1	⊢ Ⅎ𝑥𝜓
spcimgfi1.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
spcimgfi1OLD	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Proof of Theorem spcimgfi1OLD

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgfi1.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3474	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1857	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓)))
5	3, 4	biimtrid 245	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓)))
6		spcimgfi1.1	. . . 4 ⊢ Ⅎ𝑥𝜓
7	6	19.36 2268	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
8	5, 7	imbitrdi 254	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓)))
9	1, 8	syl5 35	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1561   = wceq 1563  ∃wex 1802  Ⅎwnf 1806   ∈ wcel 2145  Ⅎwnfc 2912  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-v 3459

This theorem is referenced by: (None)