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Mirrors > Home > ILE Home > Th. List > spcimegf | GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
spcimgf.1 | ⊢ Ⅎ𝑥𝐴 |
spcimgf.2 | ⊢ Ⅎ𝑥𝜓 |
spcimegf.3 | ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) |
Ref | Expression |
---|---|
spcimegf | ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcimgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | spcimgf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | spcimegft 2676 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜑)) → (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑))) |
4 | spcimegf.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
5 | 3, 4 | mpg 1380 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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: (None) |
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