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Mirrors > Home > MPE Home > Th. List > spcimgft | Structured version Visualization version GIF version |
Description: Closed theorem form of spcimgf 3550. (Contributed by Wolf Lammen, 28-Jul-2025.) |
Ref | Expression |
---|---|
spcimgft | ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elissetv 2820 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐴) | |
2 | cbvexeqsetf 3493 | . . . 4 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)) | |
3 | 1, 2 | imbitrrid 246 | . . 3 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)) |
4 | pm2.04 90 | . . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝜑 → (𝑥 = 𝐴 → 𝜓))) | |
5 | 4 | al2imi 1812 | . . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝐴 → 𝜓))) |
6 | 19.23t 2208 | . . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ (∃𝑥 𝑥 = 𝐴 → 𝜓))) | |
7 | 6 | biimpd 229 | . . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝑥 = 𝐴 → 𝜓) → (∃𝑥 𝑥 = 𝐴 → 𝜓))) |
8 | 5, 7 | sylan9r 508 | . . . 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 507 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
11 | 10 | anassrs 467 | 1 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 |
This theorem is referenced by: spcimgfi1 3547 vtoclgft 3552 |
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