![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 2195 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | sylan9req.2 | . 2 ⊢ (𝜓 → 𝐵 = 𝐶) | |
4 | 2, 3 | sylan9eq 2242 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 |
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-5 1458 ax-gen 1460 ax-4 1521 ax-17 1537 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 |
This theorem is referenced by: fndmu 5336 fodmrnu 5465 funcoeqres 5511 fvunsng 5731 prarloclem5 7530 addlocprlemeq 7563 zdiv 9372 resqrexlemnm 11062 fprodssdc 11633 dvdsmulc 11861 cncongrcoprm 12141 mgmidmo 12851 lgsmodeq 14924 |
Copyright terms: Public domain | W3C validator |