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Mirrors > Home > MPE Home > Th. List > speivw | Structured version Visualization version GIF version |
Description: Version of spei 2395 with a disjoint variable condition, which does not require ax-13 2373 (neither ax-7 2016 nor ax-12 2177). (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 251 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
3 | speivw.2 | . 2 ⊢ 𝜓 | |
4 | 2, 3 | speiv 1981 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-6 1976 |
This theorem depends on definitions: df-bi 210 df-ex 1788 |
This theorem is referenced by: elirrv 9239 bnj1014 32682 eusnsn 44225 |
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