Theorem sbcaltop 35486

Description: Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)

Assertion

Ref	Expression
sbcaltop	⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem sbcaltop

Step	Hyp	Ref	Expression
1		nfcsb1v 3913	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶
2		nfcsb1v 3913	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷
3	1, 2	nfaltop 35485	. . 3 ⊢ Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ V → Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
5		csbeq1a 3902	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
6		altopeq1 35468	. . . 4 ⊢ (𝐶 = ⦋𝐴 / 𝑥⦌𝐶 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫)
7	5, 6	syl 17	. . 3 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫)
8		csbeq1a 3902	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
9		altopeq2 35469	. . . 4 ⊢ (𝐷 = ⦋𝐴 / 𝑥⦌𝐷 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
10	8, 9	syl 17	. . 3 ⊢ (𝑥 = 𝐴 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
11	7, 10	eqtrd 2766	. 2 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
12	4, 11	csbiegf 3922	1 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877  Vcvv 3468  ⦋csb 3888  ⟪caltop 35461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-pr 4626  df-altop 35463

This theorem is referenced by: (None)