Theorem sbcbrg 4686

Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
sbcbrg	⊢ (A ∈ D → ([̣A / x]̣BRC ↔ [A / x]B[A / x]R[A / x]C))

Proof of Theorem sbcbrg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. 2 ⊢ (y = A → ([y / x]BRC ↔ [̣A / x]̣BRC))
2		csbeq1 3140	. . 3 ⊢ (y = A → [y / x]B = [A / x]B)
3		csbeq1 3140	. . 3 ⊢ (y = A → [y / x]R = [A / x]R)
4		csbeq1 3140	. . 3 ⊢ (y = A → [y / x]C = [A / x]C)
5	2, 3, 4	breq123d 4654	. 2 ⊢ (y = A → ([y / x]B[y / x]R[y / x]C ↔ [A / x]B[A / x]R[A / x]C))
6		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]B
7		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]R
8		nfcsb1v 3169	. . . 4 ⊢ Ⅎx[y / x]C
9	6, 7, 8	nfbr 4684	. . 3 ⊢ Ⅎx[y / x]B[y / x]R[y / x]C
10		csbeq1a 3145	. . . 4 ⊢ (x = y → B = [y / x]B)
11		csbeq1a 3145	. . . 4 ⊢ (x = y → R = [y / x]R)
12		csbeq1a 3145	. . . 4 ⊢ (x = y → C = [y / x]C)
13	10, 11, 12	breq123d 4654	. . 3 ⊢ (x = y → (BRC ↔ [y / x]B[y / x]R[y / x]C))
14	9, 13	sbie 2038	. 2 ⊢ ([y / x]BRC ↔ [y / x]B[y / x]R[y / x]C)
15	1, 5, 14	vtoclbg 2916	1 ⊢ (A ∈ D → ([̣A / x]̣BRC ↔ [A / x]B[A / x]R[A / x]C))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047  [csb 3137   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-csb 3138  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:  sbcbr12g  4687  csbfv12g  5337