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Mirrors > Home > ILE Home > Th. List > fvoveq1d | GIF version |
Description: Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.) |
Ref | Expression |
---|---|
fvoveq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fvoveq1d | ⊢ (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | oveq1d 5782 | . 2 ⊢ (𝜑 → (𝐴𝑂𝐶) = (𝐵𝑂𝐶)) |
3 | 2 | fveq2d 5418 | 1 ⊢ (𝜑 → (𝐹‘(𝐴𝑂𝐶)) = (𝐹‘(𝐵𝑂𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ‘cfv 5118 (class class class)co 5767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: fvoveq1 5790 imbrov2fvoveq 5792 seqvalcd 10225 mulc1cncf 12734 mulcncflem 12748 mulcncf 12749 limccl 12786 ellimc3apf 12787 limcdifap 12789 limcmpted 12790 limcresi 12793 limccoap 12805 dveflem 12844 |
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