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Mirrors > Home > MPE Home > Th. List > opabbii | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
opabbii | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ 𝑧 = 𝑧 | |
2 | opabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓)) |
4 | 3 | opabbidv 5134 | . 2 ⊢ (𝑧 = 𝑧 → {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓}) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {〈𝑥, 𝑦〉 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 {copab 5130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-opab 5131 |
This theorem is referenced by: mptv 5173 2rbropap 5453 dfid4 5463 fconstmpt 5616 xpundi 5622 xpundir 5623 inxp 5705 csbcnv 5756 cnvco 5758 resopab 5904 opabresid 5919 opabresidOLD 5921 cnvi 6002 cnvun 6003 cnvxp 6016 cnvcnv3 6047 coundi 6102 coundir 6103 mptun 6496 fvopab6 6803 fmptsng 6932 fmptsnd 6933 cbvoprab1 7243 cbvoprab12 7245 dmoprabss 7258 mpomptx 7267 resoprab 7272 elrnmpores 7290 ov6g 7314 1st2val 7719 2nd2val 7720 dfoprab3s 7753 dfoprab3 7754 dfoprab4 7755 opabn1stprc 7758 mptmpoopabbrd 7780 fsplit 7814 fsplitOLD 7815 mapsncnv 8459 xpcomco 8609 marypha2lem2 8902 oemapso 9147 leweon 9439 r0weon 9440 compsscnv 9795 fpwwe 10070 ltrelxr 10704 ltxrlt 10713 ltxr 12513 shftidt2 14442 prdsle 16737 prdsless 16738 prdsleval 16752 dfiso2 17044 joindm 17615 meetdm 17629 gaorb 18439 efgcpbllema 18882 frgpuplem 18900 ltbval 20254 ltbwe 20255 opsrle 20258 opsrtoslem1 20266 opsrtoslem2 20267 dvdsrzring 20632 pjfval2 20855 lmfval 21842 lmbr 21868 lgsquadlem3 25960 perpln1 26498 outpasch 26543 ishpg 26547 axcontlem2 26753 wksfval 27393 wlkson 27440 pthsfval 27504 ispth 27506 dfadj2 29664 dmadjss 29666 cnvadj 29671 mpomptxf 30427 satfv0 32607 satfvsuclem1 32608 satfvsuclem2 32609 satfbrsuc 32615 satf0 32621 satf0suclem 32624 fmlasuc0 32633 fneer 33703 bj-dfmpoa 34412 bj-mpomptALT 34413 bj-brab2a1 34443 opropabco 35001 cnvepres 35557 inxp2 35621 xrninxp 35642 xrninxp2 35643 rnxrnres 35649 rnxrncnvepres 35650 rnxrnidres 35651 dfcoss2 35663 dfcoss3 35664 1cossres 35676 dfcoels 35677 br1cosscnvxrn 35716 1cosscnvxrn 35717 coss0 35721 cossid 35722 dfssr2 35741 cmtfvalN 36348 cmtvalN 36349 cvrfval 36406 cvrval 36407 dicval2 38317 fgraphopab 39817 fgraphxp 39818 mptssid 41518 dfnelbr2 43479 opabbrfex0d 43492 opabbrfexd 43494 upwlksfval 44017 xpsnopab 44039 mpomptx2 44390 |
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