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Mirrors > Home > ILE Home > Th. List > equequ1 | GIF version |
Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
equequ1 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-8 1395 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | equtr 1595 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 1, 2 | impbid 120 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-gen 1338 ax-ie2 1383 ax-8 1395 ax-17 1419 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: equveli 1642 drsb1 1680 equsb3lem 1824 euequ1 1995 axext3 2023 reu6 2730 reu7 2736 cbviota 4872 dff13f 5409 poxp 5853 |
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