Theorem caovord3 5631

Description: Ordering law. (Contributed by set.mm contributors, 29-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	⊢ A ∈ V
caovord.2	⊢ B ∈ V
caovord.3	⊢ (z ∈ S → (xRy ↔ (zFx)R(zFy)))
caovord2.3	⊢ C ∈ V
caovord2.com	⊢ (xFy) = (yFx)
caovord3.4	⊢ D ∈ V

Assertion

Ref	Expression
caovord3	⊢ (((B ∈ S ∧ C ∈ S) ∧ (AFB) = (CFD)) → (ARC ↔ DRB))

Distinct variable groups:   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z   x,C,y,z   x,D,y,z   x,R,y,z

Proof of Theorem caovord3

Step	Hyp	Ref	Expression
1		caovord.1	. . . . 5 ⊢ A ∈ V
2		caovord2.3	. . . . 5 ⊢ C ∈ V
3		caovord.3	. . . . 5 ⊢ (z ∈ S → (xRy ↔ (zFx)R(zFy)))
4		caovord.2	. . . . 5 ⊢ B ∈ V
5		caovord2.com	. . . . 5 ⊢ (xFy) = (yFx)
6	1, 2, 3, 4, 5	caovord2 5630	. . . 4 ⊢ (B ∈ S → (ARC ↔ (AFB)R(CFB)))
7	6	adantr 451	. . 3 ⊢ ((B ∈ S ∧ C ∈ S) → (ARC ↔ (AFB)R(CFB)))
8		breq1 4642	. . 3 ⊢ ((AFB) = (CFD) → ((AFB)R(CFB) ↔ (CFD)R(CFB)))
9	7, 8	sylan9bb 680	. 2 ⊢ (((B ∈ S ∧ C ∈ S) ∧ (AFB) = (CFD)) → (ARC ↔ (CFD)R(CFB)))
10		caovord3.4	. . . 4 ⊢ D ∈ V
11	10, 4, 3	caovord 5629	. . 3 ⊢ (C ∈ S → (DRB ↔ (CFD)R(CFB)))
12	11	ad2antlr 707	. 2 ⊢ (((B ∈ S ∧ C ∈ S) ∧ (AFB) = (CFD)) → (DRB ↔ (CFD)R(CFB)))
13	9, 12	bitr4d 247	1 ⊢ (((B ∈ S ∧ C ∈ S) ∧ (AFB) = (CFD)) → (ARC ↔ DRB))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639  (class class class)co 5525

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526

This theorem is referenced by: (None)