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Mirrors > Home > MPE Home > Th. List > spcgft | Structured version Visualization version GIF version |
Description: A closed version of spcgf 3591. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimgfi1.1 | ⊢ Ⅎ𝑥𝜓 |
spcimgfi1.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
spcgft | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 215 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | imim2i 16 | . . 3 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓))) |
3 | 2 | alimi 1808 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) |
4 | spcimgfi1.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | spcimgfi1.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | spcimgfi1 3547 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
7 | 3, 6 | syl 17 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀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: spcgf 3591 rspct 3608 |
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