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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 7170 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐶𝐹𝐷) |
4 | oveq123i.3 | . . 3 ⊢ 𝐹 = 𝐺 | |
5 | 4 | oveqi 7171 | . 2 ⊢ (𝐶𝐹𝐷) = (𝐶𝐺𝐷) |
6 | 3, 5 | eqtri 2846 | 1 ⊢ (𝐴𝐹𝐵) = (𝐶𝐺𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: relowlpssretop 34647 mendvscafval 39797 cytpval 39816 |
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