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Mirrors > Home > MPE Home > Th. List > opabbidv | Structured version Visualization version GIF version |
Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction form). (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opabbidv | ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opabbidv.1 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | anbi2d 630 | . . . 4 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒))) |
3 | 2 | 2exbidv 1928 | . . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒))) |
4 | 3 | abbidv 2802 | . 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)}) |
5 | df-opab 5212 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)} | |
6 | df-opab 5212 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜒)} | |
7 | 4, 5, 6 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 {cab 2710 ⟨cop 4635 {copab 5211 |
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-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-opab 5212 |
This theorem is referenced by: opabbii 5216 mpteq12dva 5238 csbopab 5556 csbopabgALT 5557 csbmpt12 5558 xpeq1 5691 xpeq2 5698 opabbi2dv 5850 csbcnvgALT 5885 resopab2 6037 mptcnv 6140 cores 6249 xpco 6289 dffn5 6951 f1oiso2 7349 fvmptopab 7463 fvmptopabOLD 7464 f1ocnvd 7657 ofreq 7674 mptmpoopabbrd 8067 mptmpoopabbrdOLD 8069 bropopvvv 8076 bropfvvvv 8078 fnwelem 8117 sprmpod 8209 mpocurryd 8254 wemapwe 9692 ttrcleq 9704 xpcogend 14921 shftfval 15017 2shfti 15027 prdsval 17401 pwsle 17438 sectffval 17697 sectfval 17698 isfunc 17814 isfull 17861 isfth 17865 ipoval 18483 eqgfval 19056 eqg0subg 19073 dvdsrval 20175 dvdsrpropd 20230 ltbval 21598 opsrval 21601 lmfval 22736 xkocnv 23318 tgphaus 23621 isphtpc 24510 bcthlem1 24841 bcth 24846 ulmval 25892 lgsquadlem3 26885 iscgrg 27763 legval 27835 ishlg 27853 perpln1 27961 perpln2 27962 isperp 27963 ishpg 28010 iscgra 28060 isinag 28089 isleag 28098 wksfval 28866 upgrtrls 28958 upgrspthswlk 28995 ajfval 30062 f1o3d 31851 f1od2 31946 mgcoval 32156 inftmrel 32326 isinftm 32327 quslsm 32516 idlsrgval 32617 metidval 32870 faeval 33244 eulerpartlemgvv 33375 eulerpart 33381 afsval 33683 satf 34344 satfvsuc 34352 satfv1 34354 satf0suc 34367 sat1el2xp 34370 fmlasuc0 34375 gg-dvcnp2 35174 gg-dvmulbr 35175 gg-dvcobr 35176 gg-cmvth 35181 gg-dvfsumle 35182 gg-dvfsumlem2 35183 bj-imdirvallem 36061 bj-imdirval2 36064 bj-imdirco 36071 bj-iminvval2 36075 cureq 36464 curf 36466 curunc 36470 fnopabeqd 36589 cosseq 37296 lcvfbr 37890 cmtfvalN 38080 cvrfval 38138 dicffval 40045 dicfval 40046 dicval 40047 prjspval 41345 prjspnerlem 41359 0prjspn 41370 dnwech 41790 aomclem8 41803 tfsconcatun 42087 tfsconcat0i 42095 tfsconcatrev 42098 rfovcnvfvd 42758 fsovrfovd 42760 dfafn5a 45868 sprsymrelfv 46162 sprsymrelfo 46165 upwlksfval 46513 |
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