Theorem oveq123d 5874

Description: Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)

Hypotheses

Ref	Expression
oveq123d.1	⊢ (𝜑 → 𝐹 = 𝐺)
oveq123d.2	⊢ (𝜑 → 𝐴 = 𝐵)
oveq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
oveq123d	⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))

Proof of Theorem oveq123d

Step	Hyp	Ref	Expression
1		oveq123d.1	. . 3 ⊢ (𝜑 → 𝐹 = 𝐺)
2	1	oveqd 5870	. 2 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐴𝐺𝐶))
3		oveq123d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4		oveq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
5	3, 4	oveq12d 5871	. 2 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐵𝐺𝐷))
6	2, 5	eqtrd 2203	1 ⊢ (𝜑 → (𝐴𝐹𝐶) = (𝐵𝐺𝐷))

Syntax hints:   → wi 4   = wceq 1348  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  csbov123g  5891  issgrp  12644  sgrp1  12651  ismndd  12673  grpsubfvalg  12748