Theorem 3gencl 2889

Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
3gencl.1	⊢ (D ∈ S ↔ ∃x ∈ R A = D)
3gencl.2	⊢ (F ∈ S ↔ ∃y ∈ R B = F)
3gencl.3	⊢ (G ∈ S ↔ ∃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 groups:   x,y,z   y,D,z   z,F   x,R,y   y,S,z   ψ,x   χ,y   θ,z

Allowed substitution hints:   φ(x,y,z)   ψ(y,z)   χ(x,z)   θ(x,y)   A(x,y,z)   B(x,y,z)   C(x,y,z)   D(x)   R(z)   S(x)   F(x,y)   G(x,y,z)

Proof of Theorem 3gencl

Step	Hyp	Ref	Expression
1		3gencl.3	. . . . 5 ⊢ (G ∈ S ↔ ∃z ∈ R C = G)
2		df-rex 2620	. . . . 5 ⊢ (∃z ∈ R C = G ↔ ∃z(z ∈ R ∧ C = G))
3	1, 2	bitri 240	. . . 4 ⊢ (G ∈ S ↔ ∃z(z ∈ R ∧ C = G))
4		3gencl.6	. . . . 5 ⊢ (C = G → (χ ↔ θ))
5	4	imbi2d 307	. . . 4 ⊢ (C = G → (((D ∈ S ∧ F ∈ S) → χ) ↔ ((D ∈ S ∧ F ∈ S) → θ)))
6		3gencl.1	. . . . . 6 ⊢ (D ∈ S ↔ ∃x ∈ R A = D)
7		3gencl.2	. . . . . 6 ⊢ (F ∈ S ↔ ∃y ∈ R B = F)
8		3gencl.4	. . . . . . 7 ⊢ (A = D → (φ ↔ ψ))
9	8	imbi2d 307	. . . . . 6 ⊢ (A = D → ((z ∈ R → φ) ↔ (z ∈ R → ψ)))
10		3gencl.5	. . . . . . 7 ⊢ (B = F → (ψ ↔ χ))
11	10	imbi2d 307	. . . . . 6 ⊢ (B = F → ((z ∈ R → ψ) ↔ (z ∈ R → χ)))
12		3gencl.7	. . . . . . 7 ⊢ ((x ∈ R ∧ y ∈ R ∧ z ∈ R) → φ)
13	12	3expia 1153	. . . . . 6 ⊢ ((x ∈ R ∧ y ∈ R) → (z ∈ R → φ))
14	6, 7, 9, 11, 13	2gencl 2888	. . . . 5 ⊢ ((D ∈ S ∧ F ∈ S) → (z ∈ R → χ))
15	14	com12 27	. . . 4 ⊢ (z ∈ R → ((D ∈ S ∧ F ∈ S) → χ))
16	3, 5, 15	gencl 2887	. . 3 ⊢ (G ∈ S → ((D ∈ S ∧ F ∈ S) → θ))
17	16	com12 27	. 2 ⊢ ((D ∈ S ∧ F ∈ S) → (G ∈ S → θ))
18	17	3impia 1148	1 ⊢ ((D ∈ S ∧ F ∈ S ∧ G ∈ S) → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-rex 2620

This theorem is referenced by: (None)