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Mirrors > Home > MPE Home > Th. List > spcimgfi1 | Structured version Visualization version GIF version |
Description: A closed version of spcimgf 3550. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 27-Jul-2025.) |
Ref | Expression |
---|---|
spcimgfi1.1 | ⊢ Ⅎ𝑥𝜓 |
spcimgfi1.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
spcimgfi1 | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimgfi1.2 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | spcimgfi1.1 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | spcimgft 3546 | . . 3 ⊢ (((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) | |
4 | 3 | ex 412 | . 2 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) → (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))) |
5 | 1, 2, 4 | mp2an 692 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 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: spcgft 3549 spcimgf 3550 ss2iundf 43649 spcdvw 48910 |
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