Theorem sbcbrg 4083

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 2988	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ [𝐴 / 𝑥]𝐵𝑅𝐶))
2		csbeq1 3083	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3083	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
4		csbeq1 3083	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	breq123d 4043	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))
6		nfcsb1v 3113	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
7		nfcsb1v 3113	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅
8		nfcsb1v 3113	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	6, 7, 8	nfbr 4075	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶
10		csbeq1a 3089	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3089	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅)
12		csbeq1a 3089	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
13	10, 11, 12	breq123d 4043	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶))
14	9, 13	sbie 1802	. 2 ⊢ ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶)
15	1, 5, 14	vtoclbg 2821	1 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  [wsb 1773   ∈ wcel 2164  [wsbc 2985  ⦋csb 3080   class class class wbr 4029

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030

This theorem is referenced by:  sbcbr12g  4084  csbcnvg  4846  sbcfung  5278  csbfv12g  5592