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Mirrors > Home > ILE Home > Th. List > sylan9req | GIF version |
Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.) |
Ref | Expression |
---|---|
sylan9req.1 | ⊢ (𝜑 → 𝐵 = 𝐴) |
sylan9req.2 | ⊢ (𝜓 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
sylan9req | ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylan9req.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2093 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | sylan9req.2 | . 2 ⊢ (𝜓 → 𝐵 = 𝐶) | |
4 | 2, 3 | sylan9eq 2140 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 |
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-5 1381 ax-gen 1383 ax-4 1445 ax-17 1464 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-cleq 2081 |
This theorem is referenced by: xpid11m 4658 fndmu 5115 fodmrnu 5241 funcoeqres 5284 fvunsng 5491 prarloclem5 7057 addlocprlemeq 7090 zdiv 8832 resqrexlemnm 10447 dvdsmulc 11098 cncongrcoprm 11362 |
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