Theorem 3gencl 1367

Description: Implicit substitution for class with embedded variable.

Hypotheses

Ref	Expression
3gencl.1	⊢ (D ∈ S ↔ ∃x(x ∈ R ∧ A = D))
3gencl.2	⊢ (F ∈ S ↔ ∃y(y ∈ R ∧ B = F))
3gencl.3	⊢ (G ∈ S ↔ ∃z(z ∈ R ∧ C = G))
3gencl.4	⊢ (A = D → (φ ↔ ψ))
3gencl.5	⊢ (B = F → (ψ ↔ χ))
3gencl.6	⊢ (C = G → (χ ↔ θ))
3gencl.7	⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → φ)

Assertion

Ref	Expression
3gencl	⊢ ((D ∈ S ∧ F ∈ S ∧ G ∈ S) → θ)

Distinct variable group(s):   x,y,z   y,D,z   z,F   x,R,y   y,S,z   ψ,x   χ,y   θ,z

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . . 5 ⊢ (G ∈ S ↔ ∃z(z ∈ R ∧ C = G))
2		3gencl.6	. . . . . 6 ⊢ (C = G → (χ ↔ θ))
3	2	imbi2d 464	. . . . 5 ⊢ (C = G → (((D ∈ S ∧ F ∈ S) → χ) ↔ ((D ∈ S ∧ F ∈ S) → θ)))
4		3gencl.1	. . . . . . 7 ⊢ (D ∈ S ↔ ∃x(x ∈ R ∧ A = D))
5		3gencl.2	. . . . . . 7 ⊢ (F ∈ S ↔ ∃y(y ∈ R ∧ B = F))
6		3gencl.4	. . . . . . . 8 ⊢ (A = D → (φ ↔ ψ))
7	6	imbi2d 464	. . . . . . 7 ⊢ (A = D → ((z ∈ R → φ) ↔ (z ∈ R → ψ)))
8		3gencl.5	. . . . . . . 8 ⊢ (B = F → (ψ ↔ χ))
9	8	imbi2d 464	. . . . . . 7 ⊢ (B = F → ((z ∈ R → ψ) ↔ (z ∈ R → χ)))
10		3gencl.7	. . . . . . . . 9 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → φ)
11	10	3exp 611	. . . . . . . 8 ⊢ (x ∈ R → (y ∈ R → (z ∈ R → φ)))
12	11	imp 277	. . . . . . 7 ⊢ ((x ∈ R ∧ y ∈ R) → (z ∈ R → φ))
13	4, 5, 7, 9, 12	2gencl 1366	. . . . . 6 ⊢ ((D ∈ S ∧ F ∈ S) → (z ∈ R → χ))
14	13	com12 13	. . . . 5 ⊢ (z ∈ R → ((D ∈ S ∧ F ∈ S) → χ))
15	1, 3, 14	gencl 1365	. . . 4 ⊢ (G ∈ S → ((D ∈ S ∧ F ∈ S) → θ))
16	15	com12 13	. . 3 ⊢ ((D ∈ S ∧ F ∈ S) → (G ∈ S → θ))
17	16	exp 291	. 2 ⊢ (D ∈ S → (F ∈ S → (G ∈ S → θ)))
18	17	3imp 608	1 ⊢ ((D ∈ S ∧ F ∈ S ∧ G ∈ S) → θ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∧ w3a 581  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  ltsor 4055  axltadd 4085

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679

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