Theorem oveqan12rd 7275

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 7274	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  addpipq  10624  mulgt0sr  10792  mulcnsr  10823  mulresr  10826  recdiv  11611  revccat  14407  rlimdiv  15285  caucvg  15318  divgcdcoprm0  16298  estrchom  17759  funcestrcsetclem5  17777  ismhm  18347  xrsdsval  20554  mpfrcl  21205  matval  21468  ucnval  23337  volcn  24675  dvres2lem  24979  dvid  24987  c1lip3  25068  taylthlem1  25437  abelthlem9  25504  2sqnn  26492  brbtwn2  27176  nonbooli  29914  0cnop  30242  0cnfn  30243  idcnop  30244  bccolsum  33611  ftc1anc  35785  rmydioph  40752  expdiophlem2  40760  dvcosax  43357  ismgmhm  45225  2zrngamgm  45385  rnghmsscmap2  45419  rnghmsscmap  45420  funcrngcsetc  45444  rhmsscmap2  45465  rhmsscmap  45466  funcringcsetc  45481