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Theorem recseq 7047
Description: Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
recseq  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )

Proof of Theorem recseq
StepHypRef Expression
1 wrecseq3 6988 . 2  |-  ( F  =  G  -> wrecs (  _E  ,  On ,  F
)  = wrecs (  _E  ,  On ,  G )
)
2 df-recs 7045 . 2  |- recs ( F )  = wrecs (  _E  ,  On ,  F
)
3 df-recs 7045 . 2  |- recs ( G )  = wrecs (  _E  ,  On ,  G
)
41, 2, 33eqtr4g 2487 1  |-  ( F  =  G  -> recs ( F )  = recs ( G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    _E cep 4705   Oncon0 5385  wrecscwrecs 6982  recscrecs 7044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-xp 4802  df-cnv 4804  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-iota 5508  df-fv 5552  df-wrecs 6983  df-recs 7045
This theorem is referenced by:  rdgeq1  7084  rdgeq2  7085  dfoi  7979  oieq1  7980  oieq2  7981  ordtypecbv  7985  dfac12r  8527  zorn2g  8884  ttukey2g  8897  csbrdgg  31637  aomclem3  35827  aomclem8  35832
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