Theorem recseq 8088

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 8030	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 8086	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 8086	. 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
4	1, 2, 3	3eqtr4g 2796	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   = wceq 1543   E cep 5444  Oncon0 6191  wrecscwrecs 8024  recscrecs 8085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-iota 6316  df-fv 6366  df-wrecs 8025  df-recs 8086

This theorem is referenced by:  rdgeq1  8125  rdgeq2  8126  dfoi  9105  oieq1  9106  oieq2  9107  ordtypecbv  9111  dfac12r  9725  zorn2g  10082  ttukey2g  10095  csbrdgg  35186  aomclem3  40525  aomclem8  40530