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Mirrors > Home > MPE Home > Th. List > eqeqan12d | Structured version Visualization version GIF version |
Description: A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2754. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024.) |
Ref | Expression |
---|---|
eqeqan12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqeqan12d.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
eqeqan12d | ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeqan12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eqeq1d 2734 | . 2 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) |
3 | eqeqan12d.2 | . . 3 ⊢ (𝜓 → 𝐶 = 𝐷) | |
4 | 3 | eqeq2d 2743 | . 2 ⊢ (𝜓 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷)) |
5 | 2, 4 | sylan9bb 510 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 |
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 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 |
This theorem is referenced by: eqeqan12rd 2747 eqeq12d 2748 eqeq12 2749 eqfnfv2 7033 f1mpt 7259 soisores 7323 xpopth 8015 f1o2ndf1 8107 fnwelem 8116 fnse 8118 tz7.48lem 8440 ecopoveq 8811 xpdom2 9066 unfilem2 9310 wemaplem1 9540 suc11reg 9613 oemapval 9677 cantnf 9687 wemapwe 9691 r0weon 10006 infxpen 10008 fodomacn 10050 sornom 10271 fin1a2lem2 10395 fin1a2lem4 10397 neg11 11510 subeqrev 11635 rpnnen1lem6 12965 cnref1o 12968 xneg11 13193 injresinj 13752 modadd1 13872 modmul1 13888 modlteq 13909 sq11 14095 hashen 14306 fz1eqb 14313 eqwrd 14506 s111 14564 ccatopth 14665 wrd2ind 14672 wwlktovf1 14907 cj11 15108 sqrt11 15208 sqabs 15253 recan 15282 reeff1 16062 efieq 16105 eulerthlem2 16714 vdwlem12 16924 xpsff1o 17512 ismhm 18672 gsmsymgreq 19299 symgfixf1 19304 odf1 19429 sylow1 19470 frgpuplem 19639 isdomn 20909 cygznlem3 21124 psgnghm 21132 tgtop11 22484 fclsval 23511 vitali 25129 recosf1o 26043 dvdsmulf1o 26695 fsumvma 26713 negs11 27520 brcgr 28155 axlowdimlem15 28211 axcontlem1 28219 axcontlem4 28222 axcontlem7 28225 axcontlem8 28226 iswlk 28864 wlkswwlksf1o 29130 wwlksnextinj 29150 clwlkclwwlkf1 29260 clwwlkf1 29299 numclwwlkqhash 29625 grpoinvf 29780 hial2eq2 30355 qusker 32459 bnj554 33905 erdszelem9 34185 sategoelfvb 34405 mrsubff1 34500 msubff1 34542 mvhf1 34545 fneval 35232 topfneec2 35236 bj-imdirval3 36060 f1omptsnlem 36212 f1omptsn 36213 rdgeqoa 36246 poimirlem4 36487 poimirlem26 36509 poimirlem27 36510 ismtyval 36663 extep 37146 brdmqss 37511 wepwsolem 41774 fnwe2val 41781 aomclem8 41793 onsucf1o 42012 relexp0eq 42442 sprsymrelf1 46154 fmtnof1 46193 fmtnofac1 46228 prmdvdsfmtnof1 46245 sfprmdvdsmersenne 46261 isupwlk 46504 uspgrsprf1 46515 ismgmhm 46543 rngqiprngimfo 46776 pzriprnglem11 46805 2zlidl 46822 rrx2xpref1o 47394 rrx2plord 47396 rrx2plordisom 47399 sphere 47423 line2ylem 47427 |
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