Theorem sbcaltop 36481

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 3876	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐶
2		nfcsb1v 3876	. . . 4 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷
3	1, 2	nfaltop 36480	. . 3 ⊢ Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ V → Ⅎ𝑥⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
5		csbeq1a 3866	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐶 = ⦋𝐴 / 𝑥⦌𝐶)
6		altopeq1 36463	. . . 4 ⊢ (𝐶 = ⦋𝐴 / 𝑥⦌𝐶 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫)
7	5, 6	syl 18	. . 3 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫)
8		csbeq1a 3866	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷)
9		altopeq2 36464	. . . 4 ⊢ (𝐷 = ⦋𝐴 / 𝑥⦌𝐷 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
10	8, 9	syl 18	. . 3 ⊢ (𝑥 = 𝐴 → ⟪⦋𝐴 / 𝑥⦌𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
11	7, 10	eqtrd 2797	. 2 ⊢ (𝑥 = 𝐴 → ⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)
12	4, 11	csbiegf 3885	1 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌⟪𝐶, 𝐷⟫ = ⟪⦋𝐴 / 𝑥⦌𝐶, ⦋𝐴 / 𝑥⦌𝐷⟫)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Ⅎwnfc 2909  Vcvv 3454  ⦋csb 3852  ⟪caltop 36456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3456  df-sbc 3744  df-csb 3853  df-un 3909  df-ss 3921  df-sn 4589  df-pr 4591  df-altop 36458

This theorem is used by: (None)