Theorem gencl 2888

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

Hypotheses

Ref	Expression
gencl.1	⊢ (θ ↔ ∃x(χ ∧ A = B))
gencl.2	⊢ (A = B → (φ ↔ ψ))
gencl.3	⊢ (χ → φ)

Assertion

Ref	Expression
gencl	⊢ (θ → ψ)

Distinct variable group:   ψ,x

Allowed substitution hints:   φ(x)   χ(x)   θ(x)   A(x)   B(x)

Proof of Theorem gencl

Step	Hyp	Ref	Expression
1		gencl.1	. 2 ⊢ (θ ↔ ∃x(χ ∧ A = B))
2		gencl.3	. . . . 5 ⊢ (χ → φ)
3		gencl.2	. . . . 5 ⊢ (A = B → (φ ↔ ψ))
4	2, 3	syl5ib 210	. . . 4 ⊢ (A = B → (χ → ψ))
5	4	impcom 419	. . 3 ⊢ ((χ ∧ A = B) → ψ)
6	5	exlimiv 1634	. 2 ⊢ (∃x(χ ∧ A = B) → ψ)
7	1, 6	sylbi 187	1 ⊢ (θ → ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642

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

This theorem is referenced by:  2gencl  2889  3gencl  2890