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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 2014 | . 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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: equtr 2018 equeuclr 2020 spfw 2030 cbvalw 2032 alcomimw 2040 ax8 2112 elequ1 2113 ax9 2120 elequ2 2121 stdpc7 2248 sbequ12r 2250 cbvalv1 2342 cbval 2401 sb9 2522 ax9ALT 2730 sbralieALT 3357 rabrabi 3453 reurab 3710 reu8 3742 sbcco2 3818 reu8nf 3886 notabw 4319 sbcop1 5499 opeliunxp 5756 elrnmpt1 5974 elidinxp 6064 fvn0ssdmfun 7094 elabrex 7262 elabrexg 7263 riotarab 7430 tfisi 7880 tfinds2 7885 opabex3d 7989 opabex3rd 7990 opabex3 7991 xpord2indlem 8171 xpord3inddlem 8178 mpocurryd 8293 boxriin 8979 ixpiunwdom 9628 elirrv 9634 rabssnn0fi 14024 fproddivf 16020 prmodvdslcmf 17081 eqg0subg 19227 1mavmul 22570 ptbasfi 23605 elmptrab 23851 pcoass 25071 iundisj2 25598 dchrisumlema 27547 dchrisumlem2 27549 cusgrfilem2 29489 frgrncvvdeq 30338 frgr2wwlk1 30358 iundisj2f 32610 iundisj2fi 32805 bnj1014 34954 cvmsss2 35259 gonarlem 35379 ax8dfeq 35780 in-ax8 36207 ss-ax8 36208 bj-ssbid1ALT 36648 bj-cbvexw 36659 bj-sb 36670 finxpreclem6 37379 ralssiun 37390 wl-nfs1t 37518 wl-equsb4 37538 wl-euequf 37555 wl-ax11-lem8 37573 matunitlindflem1 37603 poimirlem26 37633 mblfinlem2 37645 sdclem2 37729 axc11-o 38933 evl1gprodd 42099 idomnnzgmulnz 42115 abbibw 42664 rexzrexnn0 42792 disjinfi 45135 dvnmptdivc 45894 iblsplitf 45926 vonn0ioo2 46646 vonn0icc2 46648 funressnvmo 46995 ichcircshi 47379 ichreuopeq 47398 paireqne 47436 reuopreuprim 47451 uspgrsprf1 47991 opeliun2xp 48178 |
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