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Mirrors > Home > ILE Home > Th. List > equequ2 | GIF version |
Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
equequ2 | ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtrr 1636 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦)) | |
2 | equtrr 1636 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑧 = 𝑦 → 𝑧 = 𝑥)) | |
3 | 2 | equcoms 1634 | . 2 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥)) |
4 | 1, 3 | impbid 127 | 1 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-gen 1378 ax-ie2 1423 ax-8 1435 ax-17 1459 ax-i9 1463 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: ax11v2 1741 ax11v 1748 ax11ev 1749 equs5or 1751 eujust 1943 euf 1946 mo23 1982 iotaval 4898 dffun4f 4938 dff13f 5430 supmoti 6406 isoti 6420 |
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