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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 7147 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐶𝐹𝐷) |
4 | oveq123i.3 | . . 3 ⊢ 𝐹 = 𝐺 | |
5 | 4 | oveqi 7148 | . 2 ⊢ (𝐶𝐹𝐷) = (𝐶𝐺𝐷) |
6 | 3, 5 | eqtri 2821 | 1 ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 (class class class)co 7135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 |
This theorem is referenced by: relowlpssretop 34781 mendvscafval 40134 cytpval 40153 |
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