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Mirrors > Home > ILE Home > Th. List > fveq12d | GIF version |
Description: Equality deduction for function value. (Contributed by FL, 22-Dec-2008.) |
Ref | Expression |
---|---|
fveq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
fveq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
fveq12d | ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq12d.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | 1 | fveq1d 5391 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
3 | fveq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fveq2d 5393 | . 2 ⊢ (𝜑 → (𝐺‘𝐴) = (𝐺‘𝐵)) |
5 | 2, 4 | eqtrd 2150 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ‘cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 |
This theorem is referenced by: nffvd 5401 fvsng 5584 tfrlem3ag 6174 tfrlem3a 6175 tfrlemi1 6197 tfr1onlem3ag 6202 omp1eomlem 6947 seq3shft 10578 climshft2 11043 fsum3 11124 ctiunctlemfo 11879 reldvg 12744 dvfvalap 12746 |
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