Theorem recseq 7470

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

Step	Hyp	Ref	Expression
1		wrecseq3 7412	. 2 |- ( F = G -> wrecs ( _E , On , F ) = wrecs ( _E , On , G ) )
2		df-recs 7468	. 2 |- recs ( F ) = wrecs ( _E , On , F )
3		df-recs 7468	. 2 |- recs ( G ) = wrecs ( _E , On , G )
4	1, 2, 3	3eqtr4g 2681	1 |- ( F = G -> recs ( F ) = recs ( G ) )

Syntax hints:   -> wi 4   = wceq 1483   _E cep 5028   Oncon0 5723  wrecscwrecs 7406  recscrecs 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407  df-recs 7468

This theorem is referenced by:  rdgeq1  7507  rdgeq2  7508  dfoi  8416  oieq1  8417  oieq2  8418  ordtypecbv  8422  dfac12r  8968  zorn2g  9325  ttukey2g  9338  csbrdgg  33175  aomclem3  37626  aomclem8  37631