Theorem sbcbrg 3804

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 2761	. 2 ⊢ (y = A → ([y / x]B𝑅𝐶 ↔ [A / x]B𝑅𝐶))
2		csbeq1 2849	. . 3 ⊢ (y = A → ⦋y / x⦌B = ⦋A / x⦌B)
3		csbeq1 2849	. . 3 ⊢ (y = A → ⦋y / x⦌𝑅 = ⦋A / x⦌𝑅)
4		csbeq1 2849	. . 3 ⊢ (y = A → ⦋y / x⦌𝐶 = ⦋A / x⦌𝐶)
5	2, 3, 4	breq123d 3769	. 2 ⊢ (y = A → (⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶 ↔ ⦋A / x⦌B⦋A / x⦌𝑅⦋A / x⦌𝐶))
6		nfcsb1v 2876	. . . 4 ⊢ Ⅎx⦋y / x⦌B
7		nfcsb1v 2876	. . . 4 ⊢ Ⅎx⦋y / x⦌𝑅
8		nfcsb1v 2876	. . . 4 ⊢ Ⅎx⦋y / x⦌𝐶
9	6, 7, 8	nfbr 3799	. . 3 ⊢ Ⅎx⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶
10		csbeq1a 2854	. . . 4 ⊢ (x = y → B = ⦋y / x⦌B)
11		csbeq1a 2854	. . . 4 ⊢ (x = y → 𝑅 = ⦋y / x⦌𝑅)
12		csbeq1a 2854	. . . 4 ⊢ (x = y → 𝐶 = ⦋y / x⦌𝐶)
13	10, 11, 12	breq123d 3769	. . 3 ⊢ (x = y → (B𝑅𝐶 ↔ ⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶))
14	9, 13	sbie 1671	. 2 ⊢ ([y / x]B𝑅𝐶 ↔ ⦋y / x⦌B⦋y / x⦌𝑅⦋y / x⦌𝐶)
15	1, 5, 14	vtoclbg 2608	1 ⊢ (A ∈ 𝐷 → ([A / x]B𝑅𝐶 ↔ ⦋A / x⦌B⦋A / x⦌𝑅⦋A / x⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsb 1642  [wsbc 2758  ⦋csb 2846   class class class wbr 3755

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756

This theorem is referenced by:  sbcbr12g  3805  csbcnvg  4462  sbcfung  4868  csbfv12g  5152