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Mirrors > Home > MPE Home > Th. List > spei | Structured version Visualization version GIF version |
Description: Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
Ref | Expression |
---|---|
spei.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
spei.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
spei | ⊢ ∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2238 | . 2 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | spei.2 | . . 3 ⊢ 𝜓 | |
3 | spei.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | mpbiri 247 | . 2 ⊢ (𝑥 = 𝑦 → 𝜑) |
5 | 1, 4 | eximii 1754 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: elirrv 8387 bnj1014 30284 |
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