Theorem oveqan12rd 7432

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 7431	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  (class class class)co 7412

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415

This theorem is referenced by:  addpipq  10936  mulgt0sr  11104  mulcnsr  11135  mulresr  11138  recdiv  11925  revccat  14721  rlimdiv  15597  caucvg  15630  divgcdcoprm0  16607  estrchom  18083  funcestrcsetclem5  18101  ismgmhm  18622  ismhm  18708  xrsdsval  21190  mpfrcl  21868  matval  22132  ucnval  24003  volcn  25356  dvres2lem  25660  dvid  25668  c1lip3  25752  taylthlem1  26122  abelthlem9  26189  2sqnn  27179  brbtwn2  28431  nonbooli  31172  0cnop  31500  0cnfn  31501  idcnop  31502  bccolsum  35014  ftc1anc  36873  rmydioph  42056  expdiophlem2  42064  dvcosax  44941  2zrngamgm  46926  rnghmsscmap2  46960  rnghmsscmap  46961  funcrngcsetc  46985  rhmsscmap2  47006  rhmsscmap  47007  funcringcsetc  47022