Theorem oveqan12rd 6569

Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.)

Hypotheses

Ref	Expression
oveq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
opreqan12i.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
oveqan12rd	⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Proof of Theorem oveqan12rd

Step	Hyp	Ref	Expression
1		oveq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		opreqan12i.2	. . 3 ⊢ (𝜓 → 𝐶 = 𝐷)
3	1, 2	oveqan12d 6568	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 468	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  addpipq  9638  mulgt0sr  9805  mulcnsr  9836  mulresr  9839  recdiv  10610  revccat  13366  rlimdiv  14224  caucvg  14257  divgcdcoprm0  15217  estrchom  16590  funcestrcsetclem5  16607  ismhm  17160  mpfrcl  19339  xrsdsval  19609  matval  20036  ucnval  21891  volcn  23180  dvres2lem  23480  dvid  23487  c1lip3  23566  taylthlem1  23931  abelthlem9  23998  brbtwn2  25585  nonbooli  27894  0cnop  28222  0cnfn  28223  idcnop  28224  bccolsum  30878  ftc1anc  32663  rmydioph  36599  expdiophlem2  36607  dvcosax  38816  ismgmhm  41573  2zrngamgm  41729  rnghmsscmap2  41765  rnghmsscmap  41766  funcrngcsetc  41790  rhmsscmap2  41811  rhmsscmap  41812  funcringcsetc  41827