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Mirrors > Home > MPE Home > Th. List > cbvralw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbvralfw 3302 with more disjoint variable conditions. (Contributed by NM, 31-Jul-2003.) Avoid ax-13 2372. (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvralw.1 | ⊢ Ⅎ𝑦𝜑 |
cbvralw.2 | ⊢ Ⅎ𝑥𝜓 |
cbvralw.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralw | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2904 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | cbvralw.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
4 | cbvralw.2 | . 2 ⊢ Ⅎ𝑥𝜓 | |
5 | cbvralw.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 1, 2, 3, 4, 5 | cbvralfw 3302 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1786 ∀wral 3062 |
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 ax-8 2109 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-clel 2811 df-nfc 2886 df-ral 3063 |
This theorem is referenced by: cbvralsvw 3315 cbvralsvwOLD 3317 cbviin 5041 disjxun 5147 ralxpf 5847 eqfnfv2f 7037 ralrnmptw 7096 dff13f 7255 ofrfval2 7691 fmpox 8053 ovmptss 8079 cbvixp 8908 mptelixpg 8929 boxcutc 8935 xpf1o 9139 indexfi 9360 ixpiunwdom 9585 dfac8clem 10027 acni2 10041 ac6c4 10476 iundom2g 10535 uniimadomf 10540 rabssnn0fi 13951 rlim2 15440 ello1mpt 15465 o1compt 15531 fsum00 15744 iserodd 16768 pcmptdvds 16827 catpropd 17653 invfuc 17927 gsummptnn0fz 19854 gsummoncoe1 21828 gsumply1eq 21829 fiuncmp 22908 elptr2 23078 ptcld 23117 ptclsg 23119 ptcnplem 23125 cnmpt11 23167 cnmpt21 23175 ovoliunlem3 25021 ovoliun 25022 ovoliun2 25023 finiunmbl 25061 volfiniun 25064 iunmbl 25070 voliun 25071 mbfeqalem1 25158 mbfsup 25181 mbfinf 25182 mbflim 25185 itg2split 25267 itgeqa 25331 itgfsum 25344 itgabs 25352 itggt0 25361 limciun 25411 dvlipcn 25511 dvfsumlem4 25546 dvfsum2 25551 itgsubst 25566 coeeq2 25756 ulmss 25909 leibpi 26447 rlimcnp 26470 o1cxp 26479 lgamgulmlem6 26538 fsumdvdscom 26689 lgseisenlem2 26879 disjunsn 31825 bnj110 33869 bnj1529 34081 poimirlem23 36511 itgabsnc 36557 itggt0cn 36558 totbndbnd 36657 disjinfi 43891 fmptf 43942 caucvgbf 44200 climinff 44327 idlimc 44342 fnlimabslt 44395 limsupref 44401 limsupbnd1f 44402 climbddf 44403 climinf2 44423 limsupubuz 44429 climinfmpt 44431 limsupmnf 44437 limsupre2 44441 limsupmnfuz 44443 limsupre3 44449 limsupre3uz 44452 limsupreuz 44453 climuz 44460 lmbr3 44463 limsupgt 44494 liminfreuz 44519 liminflt 44521 xlimpnfxnegmnf 44530 xlimmnf 44557 xlimpnf 44558 dfxlim2 44564 cncfshift 44590 stoweidlem31 44747 iundjiun 45176 meaiunincf 45199 pimgtmnf2 45430 smfpimcc 45524 smfsup 45530 smfinflem 45533 smfinf 45534 cbvral2 45811 |
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