Theorem recseq 8313

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 8267	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 8311	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 8311	. 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 1542   E cep 5530  Oncon0 6323  wrecscwrecs 8261  recscrecs 8310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fv 6506  df-ov 7370  df-frecs 8231  df-wrecs 8262  df-recs 8311

This theorem is referenced by:  rdgeq1  8350  rdgeq2  8351  dfoi  9426  oieq1  9427  oieq2  9428  ordtypecbv  9432  dfac12r  10069  zorn2g  10425  ttukey2g  10438  csbrdgg  37645  aomclem3  43484  aomclem8  43489