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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 1482 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | equtr 1685 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 1, 2 | impbid 128 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
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-gen 1425 ax-ie2 1470 ax-8 1482 ax-17 1506 ax-i9 1510 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: equveli 1732 drsb1 1771 equsb3lem 1923 euequ1 2094 axext3 2122 reu6 2873 reu7 2879 disjiun 3924 cbviota 5093 dff13f 5671 poxp 6129 dcdifsnid 6400 supmoti 6880 isoti 6894 fsum2dlemstep 11203 ennnfonelemr 11936 ctinf 11943 |
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