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Mirrors > Home > MPE Home > Th. List > spsbe | Structured version Visualization version GIF version |
Description: A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.) |
Ref | Expression |
---|---|
spsbe | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1870 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
2 | exsimpr 1784 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥𝜑) | |
3 | 1, 2 | syl 17 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 |
This theorem is referenced by: sbft 2367 2mo 2539 bj-sbftv 31951 bj-sbfvv 31953 wl-lem-moexsb 32529 spsbce-2 37602 sb5ALT 37752 sb5ALTVD 38171 |
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