Theorem recseq 8205

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 8136	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 8202	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 8202	. 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
4	1, 2, 3	3eqtr4g 2803	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   = wceq 1539   E cep 5494  Oncon0 6266  wrecscwrecs 8127  recscrecs 8201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fv 6441  df-ov 7278  df-frecs 8097  df-wrecs 8128  df-recs 8202

This theorem is referenced by:  rdgeq1  8242  rdgeq2  8243  dfoi  9270  oieq1  9271  oieq2  9272  ordtypecbv  9276  dfac12r  9902  zorn2g  10259  ttukey2g  10272  csbrdgg  35500  aomclem3  40881  aomclem8  40886