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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 5496 | . 2 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
2 | 1 | fveq2d 5500 | 1 ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ‘cfv 5198 |
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 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 |
This theorem is referenced by: difinfsnlem 7076 ctssdclemn0 7087 cc2 7229 seq3f1olemqsum 10456 seq3f1oleml 10459 seq3f1o 10460 seq3homo 10466 seq3coll 10777 fsumf1o 11353 iserabs 11438 explecnv 11468 cvgratnnlemnexp 11487 cvgratnnlemmn 11488 fprodf1o 11551 alginv 12001 algcvg 12002 algcvga 12005 ctiunctlemu1st 12389 ctiunctlemu2nd 12390 ctiunctlemudc 12392 ctiunctlemfo 12394 subctctexmid 14034 |
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