Theorem 2gencl 2889

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

Hypotheses

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

Assertion

Ref	Expression
2gencl	⊢ ((C ∈ S ∧ D ∈ S) → χ)

Distinct variable groups:   x,y   x,R   ψ,x   y,C   y,S   χ,y

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

Proof of Theorem 2gencl

Step	Hyp	Ref	Expression
1		2gencl.2	. . . 4 ⊢ (D ∈ S ↔ ∃y ∈ R B = D)
2		df-rex 2621	. . . 4 ⊢ (∃y ∈ R B = D ↔ ∃y(y ∈ R ∧ B = D))
3	1, 2	bitri 240	. . 3 ⊢ (D ∈ S ↔ ∃y(y ∈ R ∧ B = D))
4		2gencl.4	. . . 4 ⊢ (B = D → (ψ ↔ χ))
5	4	imbi2d 307	. . 3 ⊢ (B = D → ((C ∈ S → ψ) ↔ (C ∈ S → χ)))
6		2gencl.1	. . . . . 6 ⊢ (C ∈ S ↔ ∃x ∈ R A = C)
7		df-rex 2621	. . . . . 6 ⊢ (∃x ∈ R A = C ↔ ∃x(x ∈ R ∧ A = C))
8	6, 7	bitri 240	. . . . 5 ⊢ (C ∈ S ↔ ∃x(x ∈ R ∧ A = C))
9		2gencl.3	. . . . . 6 ⊢ (A = C → (φ ↔ ψ))
10	9	imbi2d 307	. . . . 5 ⊢ (A = C → ((y ∈ R → φ) ↔ (y ∈ R → ψ)))
11		2gencl.5	. . . . . 6 ⊢ ((x ∈ R ∧ y ∈ R) → φ)
12	11	ex 423	. . . . 5 ⊢ (x ∈ R → (y ∈ R → φ))
13	8, 10, 12	gencl 2888	. . . 4 ⊢ (C ∈ S → (y ∈ R → ψ))
14	13	com12 27	. . 3 ⊢ (y ∈ R → (C ∈ S → ψ))
15	3, 5, 14	gencl 2888	. 2 ⊢ (D ∈ S → (C ∈ S → χ))
16	15	impcom 419	1 ⊢ ((C ∈ S ∧ D ∈ S) → χ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616

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-ex 1542  df-rex 2621

This theorem is referenced by:  3gencl  2890