Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > equcoms | GIF version |
Description: An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
equcoms.1 | ⊢ (𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
equcoms | ⊢ (𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcomi 1680 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
2 | equcoms.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 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: equtr 1685 equtr2 1687 equequ2 1689 elequ1 1690 elequ2 1691 ax10o 1693 cbvalh 1726 cbvexh 1728 equvini 1731 stdpc7 1743 sbequ12r 1745 sbequ12a 1746 sbequ 1812 sb6rf 1825 sb9v 1953 sb6a 1963 mo2n 2027 cleqh 2239 cbvab 2263 sbralie 2670 reu8 2880 sbcco2 2931 snnex 4369 tfisi 4501 opeliunxp 4594 elrnmpt1 4790 rnxpid 4973 iotaval 5099 elabrex 5659 opabex3d 6019 opabex3 6020 enq0ref 7241 setindis 13165 bdsetindis 13167 |
Copyright terms: Public domain | W3C validator |