Theorem oveqan12rd 7440

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 7439	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 464	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  (class class class)co 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  addpipq  11022  mulgt0sr  11190  mulcnsr  11221  mulresr  11224  recdiv  12023  revccat  14915  rlimdiv  15813  caucvg  15846  divgcdcoprm0  16840  estrchom  18301  funcestrcsetclem5  18318  ismgmhm  18885  ismhm  18980  rnghmsscmap2  20881  rnghmsscmap  20882  funcrngcsetc  20892  rhmsscmap2  20910  rhmsscmap  20911  funcringcsetc  20926  xrsdsval  21717  mpfrcl  22394  matval  22726  ucnval  24595  volcn  25927  dvres2lem  26230  dvid  26238  c1lip3  26319  taylthlem1  26700  abelthlem9  26767  2sqnn  27766  brbtwn2  29483  nonbooli  32253  0cnop  32581  0cnfn  32582  idcnop  32583  bccolsum  36504  ftc1anc  38620  rmydioph  44021  expdiophlem2  44029  dvcosax  46936  2zrngamgm  49342