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Theorem 2fveq3 5499
Description: Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
Assertion
Ref Expression
2fveq3  |-  ( A  =  B  ->  ( F `  ( G `  A ) )  =  ( F `  ( G `  B )
) )

Proof of Theorem 2fveq3
StepHypRef Expression
1 fveq2 5494 . 2  |-  ( A  =  B  ->  ( G `  A )  =  ( G `  B ) )
21fveq2d 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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