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Mirrors > Home > NFE Home > Th. List > preq2 | GIF version |
Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
preq2 | ⊢ (A = B → {C, A} = {C, B}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preq1 3799 | . 2 ⊢ (A = B → {A, C} = {B, C}) | |
2 | prcom 3798 | . 2 ⊢ {C, A} = {A, C} | |
3 | prcom 3798 | . 2 ⊢ {C, B} = {B, C} | |
4 | 1, 2, 3 | 3eqtr4g 2410 | 1 ⊢ (A = B → {C, A} = {C, B}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 |
This theorem is referenced by: preq12 3801 preq2i 3803 preq2d 3806 tpeq2 3809 uniprg 3906 intprg 3960 opkeq2 4060 preqr2g 4126 preq12bg 4128 enprmapc 6083 |
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