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Mirrors > Home > MPE Home > Th. List > spcimgft | Structured version Visualization version GIF version |
Description: A closed version of spcimgf 3580. (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 3492 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | spcimgft.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3488 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | exim 1835 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓))) | |
5 | 3, 4 | biimtrid 241 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓))) |
6 | spcimgft.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
7 | 6 | 19.36 2222 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
8 | 5, 7 | imbitrdi 250 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2105 Ⅎwnfc 2882 Vcvv 3473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-v 3475 |
This theorem is referenced by: spcgft 3579 spcimgf 3580 ss2iundf 42713 spcdvw 47813 |
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