Theorem spcimgft 3515

Description: Closed theorem form of spcimgf 3519. (Contributed by Wolf Lammen, 28-Jul-2025.)

Assertion

Ref	Expression
spcimgft	⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))

Proof of Theorem spcimgft

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elissetv 2844	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴)
2		cbvexeqsetf 3470	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
3	1, 2	imbitrrid 248	. . 3 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴))
4		pm2.04 90	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝜑 → (𝑥 = 𝐴 → 𝜓)))
5	4	al2imi 1836	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜓)))
6		19.23t 2246	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓)))
7	6	biimpd 231	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
8	5, 7	sylan9r 516	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → (∃𝑥 𝑥 = 𝐴 → 𝜓)))
9	8	com23 86	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (∃𝑥 𝑥 = 𝐴 → (∀𝑥𝜑 → 𝜓)))
10	3, 9	sylan9 515	. 2 ⊢ ((Ⅎ𝑥𝐴 ∧ (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
11	10	anassrs 471	1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1559   = wceq 1561  ∃wex 1800  Ⅎwnf 1804   ∈ wcel 2143  Ⅎwnfc 2910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-nf 1805  df-cleq 2755  df-clel 2838  df-nfc 2912

This theorem is referenced by:  spcimgfi1  3516  vtoclgft  3521