Theorem rdgeq12 8408

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 8407	. 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2		rdgeq1 8406	. 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
3	1, 2	sylan9eqr 2795	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  reccrdg 8404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5681  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-iota 6492  df-fv 6548  df-ov 7407  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405

This theorem is referenced by:  seqomeq12  8449  seqeq3  13967  satf  34282  satf0  34301  csbfinxpg  36207