Theorem recseq 8386

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 8340	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 8384	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 8384	. 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
4	1, 2, 3	3eqtr4g 2820	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   E cep 5549  Oncon0 6364  wrecscwrecs 8334  recscrecs 8383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-iota 6496  df-fv 6549  df-ov 7425  df-frecs 8304  df-wrecs 8335  df-recs 8384

This theorem is used by:  rdgeq1  8424  rdgeq2  8425  dfoi  9510  oieq1  9511  oieq2  9512  ordtypecbv  9516  dfac12r  10230  zorn2g  10586  ttukey2g  10599  csbrdgg  38359  aomclem3  44165  aomclem8  44170