Theorem breq123d 4016

Description: Equality deduction for a binary relation. (Contributed by NM, 29-Oct-2011.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
breq123d.2	⊢ (𝜑 → 𝑅 = 𝑆)
breq123d.3	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
breq123d	⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Proof of Theorem breq123d

Step	Hyp	Ref	Expression
1		breq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		breq123d.3	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
3	1, 2	breq12d 4015	. 2 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑅𝐷))
4		breq123d.2	. . 3 ⊢ (𝜑 → 𝑅 = 𝑆)
5	4	breqd 4013	. 2 ⊢ (𝜑 → (𝐵𝑅𝐷 ↔ 𝐵𝑆𝐷))
6	3, 5	bitrd 188	1 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ 𝐵𝑆𝐷))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   class class class wbr 4002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003

This theorem is referenced by:  sbcbrg  4056  fmptco  5681