Theorem oveqan12rd 7460

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 7459	. 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 1570  (class class class)co 7440

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by:  addpipq  11046  mulgt0sr  11214  mulcnsr  11245  mulresr  11248  recdiv  12047  revccat  14940  rlimdiv  15839  caucvg  15872  divgcdcoprm0  16866  estrchom  18327  funcestrcsetclem5  18344  ismgmhm  18911  ismhm  19006  rnghmsscmap2  20908  rnghmsscmap  20909  funcrngcsetc  20919  rhmsscmap2  20937  rhmsscmap  20938  funcringcsetc  20953  xrsdsval  21744  mpfrcl  22421  matval  22753  ucnval  24622  volcn  25954  dvres2lem  26257  dvid  26265  c1lip3  26346  taylthlem1  26727  abelthlem9  26794  2sqnn  27793  brbtwn2  29510  nonbooli  32280  0cnop  32608  0cnfn  32609  idcnop  32610  bccolsum  36681  ftc1anc  38810  rmydioph  44212  expdiophlem2  44220  dvcosax  47139  2zrngamgm  49545