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Mirrors > Home > MPE Home > Th. List > sbcopeq1a | Structured version Visualization version Unicode version |
Description: Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3446 that avoids the existential quantifiers of copsexg 4956). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
sbcopeq1a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . . . . 5 | |
2 | vex 3203 | . . . . 5 | |
3 | 1, 2 | op2ndd 7179 | . . . 4 |
4 | 3 | eqcomd 2628 | . . 3 |
5 | sbceq1a 3446 | . . 3 | |
6 | 4, 5 | syl 17 | . 2 |
7 | 1, 2 | op1std 7178 | . . . 4 |
8 | 7 | eqcomd 2628 | . . 3 |
9 | sbceq1a 3446 | . . 3 | |
10 | 8, 9 | syl 17 | . 2 |
11 | 6, 10 | bitr2d 269 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wsbc 3435 cop 4183 cfv 5888 c1st 7166 c2nd 7167 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: dfopab2 7222 dfoprab3s 7223 |
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