Theorem recseq 7807

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 7748	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 7805	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 7805	. 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
4	1, 2, 3	3eqtr4g 2833	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Syntax hints:   → wi 4   = wceq 1507   E cep 5309  Oncon0 6023  wrecscwrecs 7742  recscrecs 7804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-xp 5406  df-cnv 5408  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-iota 6146  df-fv 6190  df-wrecs 7743  df-recs 7805

This theorem is referenced by:  rdgeq1  7844  rdgeq2  7845  dfoi  8762  oieq1  8763  oieq2  8764  ordtypecbv  8768  dfac12r  9358  zorn2g  9715  ttukey2g  9728  csbrdgg  33987  aomclem3  38997  aomclem8  39002