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Theorem fveq1 5205
 Description: Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
fveq1 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem fveq1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 breq 3794 . . 3 (𝐹 = 𝐺 → (𝐴𝐹𝑥𝐴𝐺𝑥))
21iotabidv 4916 . 2 (𝐹 = 𝐺 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐴𝐺𝑥))
3 df-fv 4938 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
4 df-fv 4938 . 2 (𝐺𝐴) = (℩𝑥𝐴𝐺𝑥)
52, 3, 43eqtr4g 2113 1 (𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   class class class wbr 3792  ℩cio 4893  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3609  df-br 3793  df-iota 4895  df-fv 4938 This theorem is referenced by:  fveq1i  5207  fveq1d  5208  fvmptdf  5286  fvmptdv2  5288  isoeq1  5469  oveq  5546  offval  5747  ofrfval  5748  offval3  5789  smoeq  5936  recseq  5952  tfr0  5968  tfrlemiex  5976  rdgeq1  5989  rdgivallem  5999  rdg0  6005  frec0g  6014  frecsuclem3  6021  frecsuc  6022  ac6sfi  6383  1fv  9098  iseqeq3  9380  shftvalg  9665  shftval4g  9666  clim  10033
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