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Mirrors > Home > MPE Home > Th. List > spcimgft | Structured version Visualization version GIF version |
Description: A closed version of spcimgf 3536. (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 3459 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
2 | spcimgft.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | issetf 3454 | . . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) |
4 | exim 1835 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜑 → 𝜓))) | |
5 | 3, 4 | syl5bi 245 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ V → ∃𝑥(𝜑 → 𝜓))) |
6 | spcimgft.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
7 | 6 | 19.36 2230 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) |
8 | 5, 7 | syl6ib 254 | . 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 1536 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 |
This theorem is referenced by: spcgft 3535 spcimgf 3536 ss2iundf 40360 spcdvw 45209 |
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