Theorem sbcbr123 4636

Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Modified by NM, 22-Aug-2018.)

Assertion

Ref	Expression
sbcbr123	⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)

Proof of Theorem sbcbr123

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3412	. 2 ⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 → 𝐴 ∈ V)
2		br0 4631	. . . 4 ⊢ ¬ ⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶
3		csbprc 3932	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝑅 = ∅)
4	3	breqd 4594	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵∅⦋𝐴 / 𝑥⦌𝐶))
5	2, 4	mtbiri 316	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)
6	5	con4i 112	. 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶 → 𝐴 ∈ V)
7		dfsbcq2 3405	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ [𝐴 / 𝑥]𝐵𝑅𝐶))
8		csbeq1 3502	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
9		csbeq1 3502	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝑅 = ⦋𝐴 / 𝑥⦌𝑅)
10		csbeq1 3502	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
11	8, 9, 10	breq123d 4597	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))
12		nfcsb1v 3515	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
13		nfcsb1v 3515	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅
14		nfcsb1v 3515	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
15	12, 13, 14	nfbr 4629	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶
16		csbeq1a 3508	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
17		csbeq1a 3508	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅)
18		csbeq1a 3508	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
19	16, 17, 18	breq123d 4597	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶))
20	15, 19	sbie 2396	. . 3 ⊢ ([𝑦 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵⦋𝑦 / 𝑥⦌𝑅⦋𝑦 / 𝑥⦌𝐶)
21	7, 11, 20	vtoclbg 3240	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶))
22	1, 6, 21	pm5.21nii 367	1 ⊢ ([𝐴 / 𝑥]𝐵𝑅𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝑅⦋𝐴 / 𝑥⦌𝐶)

Syntax hints:  ¬ wn 3   ↔ wb 195   = wceq 1475  [wsb 1867   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874   class class class wbr 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584

This theorem is referenced by:  sbcbr  4637  sbcbr12g  4638  csbcnvgALT  5229  sbcfung  5827  csbfv12  6141  relowlpssretop  32388