Theorem csbco3g 3194

Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

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 3187	. 2 ⊢ (A ∈ V → [A / x][B / y]D = [[A / x]B / y]D)
2		elex 2868	. . . 4 ⊢ (A ∈ V → A ∈ V)
3		nfcvd 2491	. . . . 5 ⊢ (A ∈ V → ℲxC)
4		sbcco3g.1	. . . . 5 ⊢ (x = A → B = C)
5	3, 4	csbiegf 3177	. . . 4 ⊢ (A ∈ V → [A / x]B = C)
6	2, 5	syl 15	. . 3 ⊢ (A ∈ V → [A / x]B = C)
7	6	csbeq1d 3143	. 2 ⊢ (A ∈ V → [[A / x]B / y]D = [C / y]D)
8	1, 7	eqtrd 2385	1 ⊢ (A ∈ V → [A / x][B / y]D = [C / y]D)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbco3gOLD  3195