Theorem recseq 8374

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 8328	. 2 ⊢ (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2		df-recs 8372	. 2 ⊢ recs(𝐹) = wrecs( E , On, 𝐹)
3		df-recs 8372	. 2 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
4	1, 2, 3	3eqtr4g 2821	1 ⊢ (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   E cep 5550  Oncon0 6361  wrecscwrecs 8322  recscrecs 8371

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-iota 6493  df-fv 6545  df-ov 7421  df-frecs 8292  df-wrecs 8323  df-recs 8372

This theorem is used by:  rdgeq1  8412  rdgeq2  8413  dfoi  9498  oieq1  9499  oieq2  9500  ordtypecbv  9504  dfac12r  10218  zorn2g  10574  ttukey2g  10587  csbrdgg  38232  aomclem3  44042  aomclem8  44047