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Mirrors > Home > ILE Home > Th. List > fveqeq2d | GIF version |
Description: Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.) |
Ref | Expression |
---|---|
fveqeq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fveqeq2d | ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | fveq2d 5425 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
3 | 2 | eqeq1d 2148 | 1 ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ‘cfv 5123 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 |
This theorem is referenced by: fveqeq2 5430 algcvga 11735 pilem3 12867 |
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