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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 5353 | . 2 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
2 | 1 | fveq2d 5357 | 1 ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ‘cfv 5059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-iota 5024 df-fv 5067 |
This theorem is referenced by: difinfsnlem 6899 ctssdclemn0 6910 seq3f1olemqsum 10114 seq3f1oleml 10117 seq3f1o 10118 seq3homo 10124 seq3coll 10426 fsumf1o 10998 iserabs 11083 explecnv 11113 cvgratnnlemnexp 11132 cvgratnnlemmn 11133 alginv 11521 algcvg 11522 algcvga 11525 |
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