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Mirrors > Home > MPE Home > Th. List > preqr2 | Structured version Visualization version Unicode version |
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 15-Jul-1993.) |
Ref | Expression |
---|---|
preqr1.a | |
preqr1.b |
Ref | Expression |
---|---|
preqr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prcom 4267 | . . 3 | |
2 | prcom 4267 | . . 3 | |
3 | 1, 2 | eqeq12i 2636 | . 2 |
4 | preqr1.a | . . 3 | |
5 | preqr1.b | . . 3 | |
6 | 4, 5 | preqr1 4379 | . 2 |
7 | 3, 6 | sylbi 207 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cvv 3200 cpr 4179 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: preq12b 4382 opth 4945 opthreg 8515 usgredgreu 26110 uspgredg2vtxeu 26112 altopthsn 32068 |
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