Theorem rdgeq12 8435

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 8434	. 2 ⊢ (𝐴 = 𝐵 → rec(𝐹, 𝐴) = rec(𝐹, 𝐵))
2		rdgeq1 8433	. 2 ⊢ (𝐹 = 𝐺 → rec(𝐹, 𝐵) = rec(𝐺, 𝐵))
3	1, 2	sylan9eqr 2791	1 ⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → rec(𝐹, 𝐴) = rec(𝐺, 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  reccrdg 8431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-xp 5671  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-iota 6494  df-fv 6549  df-ov 7416  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432

This theorem is referenced by:  seqomeq12  8476  seqeq3  14029  seqseq123d  28228  satf  35317  satf0  35336  csbfinxpg  37348