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Mirrors > Home > MPE Home > Th. List > elequ1 | Structured version Visualization version GIF version |
Description: An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.) |
Ref | Expression |
---|---|
elequ1 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax8 2113 | . 2 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) | |
2 | ax8 2113 | . . 3 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) | |
3 | 2 | equcoms 2024 | . 2 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → 𝑥 ∈ 𝑧)) |
4 | 1, 3 | impbid 211 | 1 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 |
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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: elsb1 2115 cleljust 2116 ax12wdemo 2132 cleljustALT 2362 cleljustALT2 2363 dveel1 2461 axc14 2463 sbralie 3355 unissb 4944 dftr2c 5269 axsepgfromrep 5298 nalset 5314 zfpow 5365 dtruALT2 5369 el 5438 dtruOLD 5442 zfun 7726 tz7.48lem 8441 coflton 8670 pssnn 9168 unxpdomlem1 9250 pssnnOLD 9265 zfinf 9634 aceq1 10112 aceq0 10113 aceq2 10114 dfac3 10116 dfac5lem2 10119 dfac5lem3 10120 dfac2a 10124 kmlem4 10148 zfac 10455 nd1 10582 axextnd 10586 axrepndlem1 10587 axrepndlem2 10588 axunndlem1 10590 axunnd 10591 axpowndlem2 10593 axpowndlem3 10594 axpowndlem4 10595 axregndlem1 10597 axregnd 10599 zfcndun 10610 zfcndpow 10611 zfcndinf 10613 zfcndac 10614 fpwwe2lem11 10636 axgroth3 10826 axgroth4 10827 nqereu 10924 mdetunilem9 22122 madugsum 22145 neiptopnei 22636 2ndc1stc 22955 nrmr0reg 23253 alexsubALTlem4 23554 xrsmopn 24328 itg2cn 25281 itgcn 25362 sqff1o 26686 dya2iocuni 33282 bnj849 33936 fineqvrep 34095 erdsze 34193 untsucf 34679 untangtr 34683 dfon2lem3 34757 dfon2lem6 34760 dfon2lem7 34761 dfon2 34764 axextdist 34771 distel 34775 neibastop2lem 35245 bj-elequ12 35556 bj-nfeel2 35733 bj-ru0 35823 prtlem5 37730 prtlem13 37738 prtlem16 37739 ax12el 37812 pw2f1ocnv 41776 aomclem8 41803 onsupmaxb 41988 grumnud 43045 lcosslsp 47119 |
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