Theorem sbcbrg 4043

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 2958	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ [𝐴 / 𝑥]𝐵𝑅𝐶))
2		csbeq1 3052	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3		csbeq1 3052	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
4		csbeq1 3052	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
5	2, 3, 4	breq123d 4003	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))
6		nfcsb1v 3082	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
7		nfcsb1v 3082	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅
8		nfcsb1v 3082	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
9	6, 7, 8	nfbr 4035	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶
10		csbeq1a 3058	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
11		csbeq1a 3058	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅)
12		csbeq1a 3058	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
13	10, 11, 12	breq123d 4003	. . 3 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶))
14	9, 13	sbie 1784	. 2 ⊢ ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶)
15	1, 5, 14	vtoclbg 2791	1 ⊢ (𝐴 ∈ 𝐷 → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  [wsb 1755   ∈ wcel 2141  [wsbc 2955  ⦋csb 3049   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  sbcbr12g  4044  csbcnvg  4795  sbcfung  5222  csbfv12g  5532