Theorem oveqan12rd 6670

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 6669	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 469	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  addpipq  9759  mulgt0sr  9926  mulcnsr  9957  mulresr  9960  recdiv  10731  revccat  13515  rlimdiv  14376  caucvg  14409  divgcdcoprm0  15379  estrchom  16767  funcestrcsetclem5  16784  ismhm  17337  mpfrcl  19518  xrsdsval  19790  matval  20217  ucnval  22081  volcn  23374  dvres2lem  23674  dvid  23681  c1lip3  23762  taylthlem1  24127  abelthlem9  24194  brbtwn2  25785  nonbooli  28510  0cnop  28838  0cnfn  28839  idcnop  28840  bccolsum  31625  ftc1anc  33493  rmydioph  37581  expdiophlem2  37589  dvcosax  40141  ismgmhm  41783  2zrngamgm  41939  rnghmsscmap2  41973  rnghmsscmap  41974  funcrngcsetc  41998  rhmsscmap2  42019  rhmsscmap  42020  funcringcsetc  42035