Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > speivw | Structured version Visualization version GIF version |
Description: Version of spei 2411 with a disjoint variable condition, which does not require ax-13 2389 (neither ax-7 2014 nor ax-12 2176). (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
speivw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
speivw.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
speivw | ⊢ ∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | speivw.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimprd 250 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
3 | speivw.2 | . 2 ⊢ 𝜓 | |
4 | 2, 3 | speiv 1975 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-6 1969 |
This theorem depends on definitions: df-bi 209 df-ex 1780 |
This theorem is referenced by: elirrv 9053 bnj1014 32252 eusnsn 43342 |
Copyright terms: Public domain | W3C validator |