Theorem oveqan12rd 7366

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 7365	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  (class class class)co 7346

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  addpipq  10828  mulgt0sr  10996  mulcnsr  11027  mulresr  11030  recdiv  11827  revccat  14673  rlimdiv  15553  caucvg  15586  divgcdcoprm0  16576  estrchom  18033  funcestrcsetclem5  18050  ismgmhm  18604  ismhm  18693  rnghmsscmap2  20544  rnghmsscmap  20545  funcrngcsetc  20555  rhmsscmap2  20573  rhmsscmap  20574  funcringcsetc  20589  xrsdsval  21347  mpfrcl  22020  matval  22326  ucnval  24191  volcn  25534  dvres2lem  25838  dvid  25846  c1lip3  25931  taylthlem1  26308  abelthlem9  26377  2sqnn  27377  brbtwn2  28883  nonbooli  31631  0cnop  31959  0cnfn  31960  idcnop  31961  bccolsum  35783  ftc1anc  37740  rmydioph  43106  expdiophlem2  43114  dvcosax  46023  2zrngamgm  48344