Theorem sbcaltop 34210

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 3853	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶
2		nfcsb1v 3853	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷
3	1, 2	nfaltop 34209	. . 3 ⊢ Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ V → Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
5		csbeq1a 3842	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
6		altopeq1 34192	. . . 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 3842	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
9		altopeq2 34193	. . . 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 2778	. 2 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
12	4, 11	csbiegf 3862	1 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  Vcvv 3422  ⦋csb 3828  ⟪caltop 34185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561  df-altop 34187

This theorem is referenced by: (None)