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Mirrors > Home > NFE Home > Th. List > opkeq2i | GIF version |
Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006.) |
Ref | Expression |
---|---|
opkeq1i.1 | ⊢ A = B |
Ref | Expression |
---|---|
opkeq2i | ⊢ ⟪C, A⟫ = ⟪C, B⟫ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opkeq1i.1 | . 2 ⊢ A = B | |
2 | opkeq2 4060 | . 2 ⊢ (A = B → ⟪C, A⟫ = ⟪C, B⟫) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ⟪C, A⟫ = ⟪C, B⟫ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ⟪copk 4057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 df-opk 4058 |
This theorem is referenced by: 0cminle 4461 |
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