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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 5482 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
3 | fveq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | fveq2d 5484 | . 2 ⊢ (𝜑 → (𝐺‘𝐴) = (𝐺‘𝐵)) |
5 | 2, 4 | eqtrd 2197 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ‘cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 |
This theorem is referenced by: nffvd 5492 fvsng 5675 tfrlem3ag 6268 tfrlem3a 6269 tfrlemi1 6291 tfr1onlem3ag 6296 omp1eomlem 7050 seq3shft 10766 climshft2 11233 fsum3 11314 ctiunctlemfo 12315 reldvg 13195 dvfvalap 13197 |
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