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Mirrors > Home > ILE Home > Th. List > opeq2 | GIF version |
Description: Equality theorem for ordered pairs. (Contributed by NM, 25-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opeq2 | ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2240 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
2 | 1 | anbi2d 464 | . . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V))) |
3 | eqidd 2178 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐶} = {𝐶}) | |
4 | preq2 3669 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵}) | |
5 | 3, 4 | preq12d 3676 | . . . . . 6 ⊢ (𝐴 = 𝐵 → {{𝐶}, {𝐶, 𝐴}} = {{𝐶}, {𝐶, 𝐵}}) |
6 | 5 | eleq2d 2247 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ {{𝐶}, {𝐶, 𝐴}} ↔ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})) |
7 | 2, 6 | anbi12d 473 | . . . 4 ⊢ (𝐴 = 𝐵 → (((𝐶 ∈ V ∧ 𝐴 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}))) |
8 | df-3an 980 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ ((𝐶 ∈ V ∧ 𝐴 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}})) | |
9 | df-3an 980 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}) ↔ ((𝐶 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})) | |
10 | 7, 8, 9 | 3bitr4g 223 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}}) ↔ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}}))) |
11 | 10 | abbidv 2295 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ (𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}})} = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})}) |
12 | df-op 3600 | . 2 ⊢ 〈𝐶, 𝐴〉 = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐴 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐴}})} | |
13 | df-op 3600 | . 2 ⊢ 〈𝐶, 𝐵〉 = {𝑥 ∣ (𝐶 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐶}, {𝐶, 𝐵}})} | |
14 | 11, 12, 13 | 3eqtr4g 2235 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 {cab 2163 Vcvv 2737 {csn 3591 {cpr 3592 〈cop 3594 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 |
This theorem is referenced by: opeq12 3778 opeq2i 3780 opeq2d 3783 oteq2 3786 oteq3 3787 breq2 4004 cbvopab2 4074 cbvopab2v 4077 opthg 4235 eqvinop 4240 opelopabsb 4257 opelxp 4653 opabid2 4754 elrn2g 4813 opeldm 4826 opeldmg 4828 elrn2 4865 opelresg 4910 iss 4949 elimasng 4992 issref 5007 dmsnopg 5096 cnvsng 5110 elxp4 5112 elxp5 5113 dffun5r 5224 funopg 5246 f1osng 5498 tz6.12f 5540 fsn 5684 fsng 5685 fvsng 5708 oveq2 5877 cbvoprab2 5942 ovg 6007 opabex3d 6116 opabex3 6117 op1stg 6145 op2ndg 6146 oprssdmm 6166 op1steq 6174 dfoprab4f 6188 tfrlemibxssdm 6322 tfr1onlembxssdm 6338 tfrcllembxssdm 6351 elixpsn 6729 ixpsnf1o 6730 mapsnen 6805 xpsnen 6815 xpassen 6824 xpf1o 6838 djulclr 7042 djurclr 7043 djulcl 7044 djurcl 7045 djulclb 7048 inl11 7058 djuss 7063 1stinl 7067 2ndinl 7068 1stinr 7069 2ndinr 7070 elreal 7818 ax1rid 7867 fseq1p1m1 10080 cnmpt21 13458 djucllem 14208 |
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