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Mirrors > Home > MPE Home > Th. List > spcimgft | Structured version Visualization version GIF version |
Description: A closed version of spcimgf 3259. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimgft.1 | ⊢ Ⅎ𝑥𝜓 |
spcimgft.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
spcimgft | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | spcimgft.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3181 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | exim 1751 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓))) | |
5 | 3, 4 | syl5bi 231 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓))) |
6 | spcimgft.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
7 | 6 | 19.36 2085 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
8 | 5, 7 | syl6ib 240 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓))) |
9 | 1, 8 | syl5 33 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∃wex 1695 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: spcgft 3258 spcimgf 3259 spcimdv 3263 ss2iundf 36970 spcdvw 42224 |
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