Theorem spcimgfi1OLD 3496

Description: Obsolete version of spcimgfi1 3495 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 3453	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		spcimgfi1.2	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	issetf 3449	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
4		exim 1841	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓)))
5	3, 4	biimtrid 243	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓)))
6		spcimgfi1.1	. . . 4 ⊢ Ⅎ𝑥𝜓
7	6	19.36 2242	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
8	5, 7	imbitrdi 252	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓)))
9	1, 8	syl5 34	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4  ∀wal 1545   = wceq 1547  ∃wex 1786  Ⅎwnf 1790   ∈ wcel 2119  Ⅎwnfc 2887  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-v 3434

This theorem is referenced by: (None)