Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > speiv | GIF version |
Description: Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.) |
Ref | Expression |
---|---|
speiv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
speiv.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
speiv | ⊢ ∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | speiv.2 | . 2 ⊢ 𝜓 | |
2 | speiv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimprd 147 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
4 | 3 | spimev 1741 | . 2 ⊢ (𝜓 → ∃𝑥𝜑) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |