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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 5480 | . 2 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
2 | 1 | fveq2d 5484 | 1 ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(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: difinfsnlem 7055 ctssdclemn0 7066 cc2 7199 seq3f1olemqsum 10425 seq3f1oleml 10428 seq3f1o 10429 seq3homo 10435 seq3coll 10741 fsumf1o 11317 iserabs 11402 explecnv 11432 cvgratnnlemnexp 11451 cvgratnnlemmn 11452 fprodf1o 11515 alginv 11958 algcvg 11959 algcvga 11962 ctiunctlemu1st 12310 ctiunctlemu2nd 12311 ctiunctlemudc 12313 ctiunctlemfo 12315 subctctexmid 13722 |
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