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Mirrors > Home > MPE Home > Th. List > rexeqbidv | Structured version Visualization version GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Ref | Expression |
---|---|
raleqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
raleqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexeqbidv | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | rexeqdv 3122 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓)) |
3 | raleqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 3 | rexbidv 3034 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | bitrd 267 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 |
This theorem is referenced by: supeq123d 8239 fpwwe2lem13 9343 hashge2el2difr 13118 vdwpc 15522 ramval 15550 mreexexlemd 16127 iscat 16156 iscatd 16157 catidex 16158 gsumval2a 17102 ismnddef 17119 mndpropd 17139 isgrp 17251 isgrpd2e 17264 cayleyth 17658 psgnfval 17743 iscyg 18104 ltbval 19292 opsrval 19295 scmatval 20129 pmatcollpw3fi1lem2 20411 pmatcollpw3fi1 20412 neiptopnei 20746 is1stc 21054 2ndc1stc 21064 2ndcsep 21072 islly 21081 isnlly 21082 ucnval 21891 imasdsf1olem 21988 met2ndc 22138 evthicc 23035 istrkgld 25158 legval 25279 ishpg 25451 iscgra 25501 isinag 25529 isleag 25533 nb3graprlem2 25981 erclwwlkeq 26339 2wlksot 26394 2spthsot 26395 isplig 26720 nmoofval 27001 istrkg2d 29997 iscvm 30495 cvmlift2lem13 30551 br8 30899 br6 30900 br4 30901 brsegle 31385 hilbert1.1 31431 poimirlem26 32605 poimirlem28 32607 poimirlem29 32608 cover2g 32679 isexid 32816 isrngo 32866 isrngod 32867 isgrpda 32924 lshpset 33283 cvrfval 33573 isatl 33604 ishlat1 33657 llnset 33809 lplnset 33833 lvolset 33876 lineset 34042 lcfl7N 35808 lcfrlem8 35856 lcfrlem9 35857 lcf1o 35858 hvmapffval 36065 hvmapfval 36066 hvmapval 36067 mzpcompact2lem 36332 eldioph 36339 aomclem8 36649 clsk1independent 37364 ovnval 39431 nnsum3primes4 40204 nnsum3primesprm 40206 nnsum3primesgbe 40208 wtgoldbnnsum4prm 40218 bgoldbnnsum3prm 40220 nbgrval 40560 nb3grprlem2 40609 1loopgrvd0 40719 erclwwlkseq 41239 eucrctshift 41411 zlidlring 41718 uzlidlring 41719 lcoop 41994 ldepsnlinc 42091 nnpw2p 42178 |
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