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Mirrors > Home > MPE Home > Th. List > opthg | Structured version Visualization version Unicode version |
Description: Ordered pair theorem. and are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4402 | . . . 4 | |
2 | 1 | eqeq1d 2624 | . . 3 |
3 | eqeq1 2626 | . . . 4 | |
4 | 3 | anbi1d 741 | . . 3 |
5 | 2, 4 | bibi12d 335 | . 2 |
6 | opeq2 4403 | . . . 4 | |
7 | 6 | eqeq1d 2624 | . . 3 |
8 | eqeq1 2626 | . . . 4 | |
9 | 8 | anbi2d 740 | . . 3 |
10 | 7, 9 | bibi12d 335 | . 2 |
11 | vex 3203 | . . 3 | |
12 | vex 3203 | . . 3 | |
13 | 11, 12 | opth 4945 | . 2 |
14 | 5, 10, 13 | vtocl2g 3270 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: opth1g 4947 opthg2 4948 opthneg 4950 otthg 4954 oteqex 4964 s111 13395 embedsetcestrclem 16797 symg2bas 17818 frgpnabllem1 18276 frgpnabllem2 18277 mat1dimbas 20278 dvheveccl 36401 hoidmv1le 40808 |
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