Theorem csbco3g 4411

Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sbcco3g.1	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
csbco3g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbco3g

Step	Hyp	Ref	Expression
1		csbnestg 4409	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷)
2		elex 3485	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
3		nfcvd 2900	. . . . 5 ⊢ (𝐴 ∈ V → Ⅎ𝑥𝐶)
4		sbcco3g.1	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5	3, 4	csbiegf 3912	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
6	2, 5	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
7	6	csbeq1d 3883	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋⦋𝐴 / 𝑥⦌𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷)
8	1, 7	eqtrd 2771	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐷)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⦋csb 3879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-v 3466  df-sbc 3771  df-csb 3880

This theorem is referenced by: (None)