![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) Remove usage of ax-10 2138, ax-11 2154, and ax-12 2174 and reduce distinct variable conditions. (Revised by Steven Nguyen, 30-Apr-2023.) |
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 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2824 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | raleqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒))) |
5 | 4 | rexbidv2 3172 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-clel 2813 df-rex 3068 |
This theorem is referenced by: frd 5644 supeq123d 9487 fpwwe2lem12 10679 vdwpc 17013 ramval 17041 mreexexlemd 17688 iscat 17716 iscatd 17717 catidex 17718 gsumval2a 18710 ismnddef 18761 mndpropd 18784 isgrp 18969 isgrpd2e 18985 cayleyth 19447 psgnfval 19532 iscyg 19911 ltbval 22078 opsrval 22081 scmatval 22525 pmatcollpw3fi1lem2 22808 pmatcollpw3fi1 22809 neiptopnei 23155 is1stc 23464 2ndc1stc 23474 2ndcsep 23482 islly 23491 isnlly 23492 ucnval 24301 imasdsf1olem 24398 met2ndc 24551 evthicc 25507 elmade2 27921 addsval 28009 mulsval 28149 istrkgb 28477 istrkge 28479 istrkgld 28481 legval 28606 ishpg 28781 iscgra 28831 isinag 28860 isleag 28869 nbgrval 29367 nb3grprlem2 29412 1loopgrvd0 29536 erclwwlkeq 30046 eucrctshift 30271 isplig 30504 nmoofval 30790 erlval 33244 idomsubr 33290 elrsp 33379 1arithidom 33544 dfufd2lem 33556 constrsuc 33742 reprsuc 34608 istrkg2d 34659 iscvm 35243 cvmlift2lem13 35299 br8 35735 br6 35736 br4 35737 brsegle 36089 hilbert1.1 36135 pibp21 37397 poimirlem26 37632 poimirlem28 37634 poimirlem29 37635 cover2g 37702 isexid 37833 isrngo 37883 isrngod 37884 isgrpda 37941 lshpset 38959 cvrfval 39249 isatl 39280 ishlat1 39333 llnset 39487 lplnset 39511 lvolset 39554 lineset 39720 lcfl7N 41483 lcfrlem8 41531 lcfrlem9 41532 lcf1o 41533 hvmapffval 41740 hvmapfval 41741 hvmapval 41742 prjspval 42589 mzpcompact2lem 42738 eldioph 42745 aomclem8 43049 tfsconcatun 43326 clsk1independent 44035 ovnval 46496 sprval 47403 nnsum3primes4 47712 nnsum3primesprm 47714 nnsum3primesgbe 47716 wtgoldbnnsum4prm 47726 bgoldbnnsum3prm 47728 clnbgrval 47746 zlidlring 48077 uzlidlring 48078 lcoop 48256 ldepsnlinc 48353 nnpw2p 48435 lines 48580 iscnrm3r 48744 |
Copyright terms: Public domain | W3C validator |