Theorem oveqan12rd 7451

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 7450	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴𝐹𝐶) = (𝐵𝐹𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  addpipq  10975  mulgt0sr  11143  mulcnsr  11174  mulresr  11177  recdiv  11971  revccat  14801  rlimdiv  15679  caucvg  15712  divgcdcoprm0  16699  estrchom  18182  funcestrcsetclem5  18200  ismgmhm  18722  ismhm  18811  rnghmsscmap2  20646  rnghmsscmap  20647  funcrngcsetc  20657  rhmsscmap2  20675  rhmsscmap  20676  funcringcsetc  20691  xrsdsval  21446  mpfrcl  22127  matval  22431  ucnval  24302  volcn  25655  dvres2lem  25960  dvid  25968  c1lip3  26053  taylthlem1  26430  abelthlem9  26499  2sqnn  27498  brbtwn2  28935  nonbooli  31680  0cnop  32008  0cnfn  32009  idcnop  32010  bccolsum  35719  ftc1anc  37688  rmydioph  43003  expdiophlem2  43011  dvcosax  45882  2zrngamgm  48089