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Mirrors > Home > ILE Home > Th. List > sb2 | GIF version |
Description: One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sb2 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-4 1487 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑)) | |
2 | equs4 1703 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | df-sb 1736 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
4 | 1, 2, 3 | sylanbrc 413 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 ∃wex 1468 [wsb 1735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 df-sb 1736 |
This theorem is referenced by: stdpc4 1748 equsb1 1758 equsb2 1759 sbiedh 1760 sb6f 1775 hbsb2a 1778 hbsb2e 1779 sbcof2 1782 sb3 1803 sb4b 1806 sb4bor 1807 hbsb2 1808 nfsb2or 1809 sb6rf 1825 sbi1v 1863 sbalyz 1972 iota4 5101 |
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