Theorem breq12d 4653

Description: Equality deduction for a binary relation. (The proof was shortened by Andrew Salmon, 9-Jul-2011.) (Contributed by NM, 8-Feb-1996.) (Revised by set.mm contributors, 9-Jul-2011.)

Hypotheses

Ref	Expression
breq1d.1	⊢ (φ → A = B)
breq12d.2	⊢ (φ → C = D)

Assertion

Ref	Expression
breq12d	⊢ (φ → (ARC ↔ BRD))

Proof of Theorem breq12d

Step	Hyp	Ref	Expression
1		breq1d.1	. 2 ⊢ (φ → A = B)
2		breq12d.2	. 2 ⊢ (φ → C = D)
3		breq12 4645	. 2 ⊢ ((A = B ∧ C = D) → (ARC ↔ BRD))
4	1, 2, 3	syl2anc 642	1 ⊢ (φ → (ARC ↔ BRD))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641

This theorem is referenced by:  breq123d  4654  3brtr3d  4669  3brtr4d  4670  isoeq1  5483  isocnv  5492  caovord  5630  refd  5928  fundmeng  6045  xpsneng  6051  xpcomeng  6054  nnltp1c  6263  nchoicelem19  6308  elcan  6330