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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 1504 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | equtr 1709 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
3 | 1, 2 | impbid 129 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
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-gen 1449 ax-ie2 1494 ax-8 1504 ax-17 1526 ax-i9 1530 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: equveli 1759 drsb1 1799 equsb3lem 1950 euequ1 2121 axext3 2160 cbvreuvw 2709 reu6 2926 reu7 2932 disjiun 3998 cbviota 5183 dff13f 5770 poxp 6232 dcdifsnid 6504 supmoti 6991 isoti 7005 nninfwlpoim 7175 exmidontriimlem3 7221 exmidontriim 7223 netap 7252 fsum2dlemstep 11437 ennnfonelemr 12418 ctinf 12425 reap0 14688 |
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