Theorem rdgeq12 8401

Description: Equality theorem for the recursive definition generator. (Contributed by Scott Fenton, 28-Apr-2012.)

Assertion

Ref	Expression
rdgeq12	⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Proof of Theorem rdgeq12

Step	Hyp	Ref	Expression
1		rdgeq2 8400	. 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2		rdgeq1 8399	. 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
3	1, 2	sylan9eqr 2820	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  reccrdg 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-iota 6494  df-fv 6546  df-ov 7415  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398

This theorem is referenced by:  seqomeq12  8442  seqeq3  14044  seqseq123d  28460  satf  35826  satf0  35845  csbfinxpg  38015