Theorem csbco3g 2036

Description: Composition of two class substitutions.

Hypothesis

Ref	Expression
csbco3g.1	⊢ (x = A → B = D)

Assertion

Ref	Expression
csbco3g	⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [A / x][B / y]C = [D / y]C)

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

Proof of Theorem csbco3g

Step	Hyp	Ref	Expression
1		csbnestg 2032	. 2 ⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [A / x][B / y]C = [[A / x]B / y]C)
2		ax-17 969	. . . . . 6 ⊢ (z ∈ D → ∀x z ∈ D)
3	2	gen2 981	. . . . 5 ⊢ ∀x∀z(z ∈ D → ∀x z ∈ D)
4		csbco3g.1	. . . . . 6 ⊢ (x = A → B = D)
5	4	ax-gen 961	. . . . 5 ⊢ ∀x(x = A → B = D)
6		csbiegft 2025	. . . . 5 ⊢ ((A ∈ R ⋀ ∀x∀z(z ∈ D → ∀x z ∈ D) ⋀ ∀x(x = A → B = D)) → [A / x]B = D)
7	3, 5, 6	mp3an23 906	. . . 4 ⊢ (A ∈ R → [A / x]B = D)
8	7	csbeq1d 2000	. . 3 ⊢ (A ∈ R → [[A / x]B / y]C = [D / y]C)
9	8	adantr 389	. 2 ⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [[A / x]B / y]C = [D / y]C)
10	1, 9	eqtrd 1504	1 ⊢ ((A ∈ R ⋀ ∀x B ∈ S) → [A / x][B / y]C = [D / y]C)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  [csb 1997

This theorem is referenced by:  fsumrev 6975  fsumshft 6977  fsum0diag2 7202

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938  df-csb 1998

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