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Mirrors > Home > ILE Home > Th. List > sb4e | GIF version |
Description: One direction of a simplified definition of substitution that unlike sb4 1832 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
sb4e | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1766 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
2 | equs5e 1795 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | |
3 | 1, 2 | syl 14 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1351 ∃wex 1492 [wsb 1762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-11 1506 ax-4 1510 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-sb 1763 |
This theorem is referenced by: hbsb2e 1807 |
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