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Mirrors > Home > ILE Home > Th. List > 2fveq3 | GIF version |
Description: Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.) |
Ref | Expression |
---|---|
2fveq3 | ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5494 | . 2 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
2 | 1 | fveq2d 5498 | 1 ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ‘cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 |
This theorem is referenced by: difinfsnlem 7072 ctssdclemn0 7083 cc2 7216 seq3f1olemqsum 10443 seq3f1oleml 10446 seq3f1o 10447 seq3homo 10453 seq3coll 10764 fsumf1o 11340 iserabs 11425 explecnv 11455 cvgratnnlemnexp 11474 cvgratnnlemmn 11475 fprodf1o 11538 alginv 11988 algcvg 11989 algcvga 11992 ctiunctlemu1st 12376 ctiunctlemu2nd 12377 ctiunctlemudc 12379 ctiunctlemfo 12381 subctctexmid 13994 |
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