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Mirrors > Home > MPE Home > Th. List > oveq123i | Structured version Visualization version GIF version |
Description: Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
Ref | Expression |
---|---|
oveq123i.1 | ⊢ 𝐴 = 𝐶 |
oveq123i.2 | ⊢ 𝐵 = 𝐷 |
oveq123i.3 | ⊢ 𝐹 = 𝐺 |
Ref | Expression |
---|---|
oveq123i | ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq123i.1 | . . 3 ⊢ 𝐴 = 𝐶 | |
2 | oveq123i.2 | . . 3 ⊢ 𝐵 = 𝐷 | |
3 | 1, 2 | oveq12i 7431 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐶𝐹𝐷) |
4 | oveq123i.3 | . . 3 ⊢ 𝐹 = 𝐺 | |
5 | 4 | oveqi 7432 | . 2 ⊢ (𝐶𝐹𝐷) = (𝐶𝐺𝐷) |
6 | 3, 5 | eqtri 2753 | 1 ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 |
This theorem is referenced by: relowlpssretop 36974 mendvscafval 42756 cytpval 42772 |
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