Theorem rdgeq12 8327

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 8326	. 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2		rdgeq1 8325	. 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
3	1, 2	sylan9eqr 2787	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  reccrdg 8323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-xp 5620  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-iota 6433  df-fv 6485  df-ov 7344  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324

This theorem is referenced by:  seqomeq12  8368  seqeq3  13905  seqseq123d  28209  satf  35365  satf0  35384  csbfinxpg  37401