![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equcoms | Structured version Visualization version GIF version |
Description: An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.) |
Ref | Expression |
---|---|
equcoms.1 | ⊢ (𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
equcoms | ⊢ (𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equcomi 2020 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
2 | equcoms.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 |
This theorem is referenced by: equtr 2024 equeuclr 2026 spfw 2036 cbvalw 2038 alcomiw 2046 ax8 2112 elequ1 2113 ax9 2120 elequ2 2121 stdpc7 2242 sbequ12r 2244 cbvalv1 2337 cbval 2397 sb9 2518 ax9ALT 2727 sbralieALT 3355 rabrabi 3450 reurab 3697 reu8 3729 sbcco2 3804 reu8nf 3871 notabw 4303 ab0OLD 4375 sbcop1 5488 opeliunxp 5743 elrnmpt1 5957 elidinxp 6043 fvn0ssdmfun 7076 elabrex 7244 riotarab 7410 tfisi 7850 tfinds2 7855 opabex3d 7954 opabex3rd 7955 opabex3 7956 xpord2indlem 8135 xpord3inddlem 8142 mpocurryd 8256 boxriin 8936 ixpiunwdom 9587 elirrv 9593 rabssnn0fi 13953 fproddivf 15933 prmodvdslcmf 16982 eqg0subg 19075 1mavmul 22057 ptbasfi 23092 elmptrab 23338 pcoass 24547 iundisj2 25073 dchrisumlema 26998 dchrisumlem2 27000 cusgrfilem2 28751 frgrncvvdeq 29600 frgr2wwlk1 29620 iundisj2f 31859 iundisj2fi 32046 bnj1014 34041 cvmsss2 34334 gonarlem 34454 ax8dfeq 34839 bj-ssbid1ALT 35628 bj-cbvexw 35639 bj-sb 35651 finxpreclem6 36363 ralssiun 36374 wl-nfs1t 36492 wl-equsb4 36508 wl-euequf 36525 wl-ax11-lem8 36540 matunitlindflem1 36570 poimirlem26 36600 mblfinlem2 36612 sdclem2 36696 axc11-o 37907 rexzrexnn0 41624 elabrexg 43810 disjinfi 43970 dvnmptdivc 44733 iblsplitf 44765 vonn0ioo2 45485 vonn0icc2 45487 funressnvmo 45834 ichcircshi 46201 ichreuopeq 46220 paireqne 46258 reuopreuprim 46273 uspgrsprf1 46604 opeliun2xp 47087 |
Copyright terms: Public domain | W3C validator |