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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 2021 | . 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 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: equtr 2025 equeuclr 2027 spfw 2037 cbvalw 2039 alcomiw 2047 ax8 2113 elequ1 2114 ax9 2121 elequ2 2122 stdpc7 2243 sbequ12r 2245 cbvalv1 2338 cbval 2398 sb9 2519 ax9ALT 2728 sbralieALT 3356 rabrabi 3451 reurab 3698 reu8 3730 sbcco2 3805 reu8nf 3872 notabw 4304 ab0OLD 4376 sbcop1 5489 opeliunxp 5744 elrnmpt1 5958 elidinxp 6044 fvn0ssdmfun 7077 elabrex 7242 riotarab 7408 tfisi 7848 tfinds2 7853 opabex3d 7952 opabex3rd 7953 opabex3 7954 xpord2indlem 8133 xpord3inddlem 8140 mpocurryd 8254 boxriin 8934 ixpiunwdom 9585 elirrv 9591 rabssnn0fi 13951 fproddivf 15931 prmodvdslcmf 16980 eqg0subg 19073 1mavmul 22050 ptbasfi 23085 elmptrab 23331 pcoass 24540 iundisj2 25066 dchrisumlema 26991 dchrisumlem2 26993 cusgrfilem2 28713 frgrncvvdeq 29562 frgr2wwlk1 29582 iundisj2f 31821 iundisj2fi 32008 bnj1014 33972 cvmsss2 34265 gonarlem 34385 ax8dfeq 34770 bj-ssbid1ALT 35542 bj-cbvexw 35553 bj-sb 35565 finxpreclem6 36277 ralssiun 36288 wl-nfs1t 36406 wl-equsb4 36422 wl-euequf 36439 wl-ax11-lem8 36454 matunitlindflem1 36484 poimirlem26 36514 mblfinlem2 36526 sdclem2 36610 axc11-o 37821 rexzrexnn0 41542 elabrexg 43728 disjinfi 43891 dvnmptdivc 44654 iblsplitf 44686 vonn0ioo2 45406 vonn0icc2 45408 funressnvmo 45755 ichcircshi 46122 ichreuopeq 46141 paireqne 46179 reuopreuprim 46194 uspgrsprf1 46525 opeliun2xp 47008 |
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