Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > spcegft | GIF version |
Description: A closed version of spcegf 2681. (Contributed by Jim Kingdon, 22-Jun-2018.) |
Ref | Expression |
---|---|
spcimgft.1 | ⊢ Ⅎ𝑥𝜓 |
spcimgft.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
spcegft | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2 128 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
2 | 1 | imim2i 12 | . . 3 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑))) |
3 | 2 | alimi 1384 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑))) |
4 | spcimgft.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | spcimgft.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | spcimegft 2676 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))) |
7 | 3, 6 | syl 14 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝐵 → (𝜓 → ∃𝑥𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∃wex 1421 ∈ wcel 1433 Ⅎwnfc 2206 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 |
This theorem is referenced by: spcegf 2681 |
Copyright terms: Public domain | W3C validator |