Theorem pm5.21nd 868

Description: Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)

Hypotheses

Ref	Expression
pm5.21nd.1	⊢ ((φ ∧ ψ) → θ)
pm5.21nd.2	⊢ ((φ ∧ χ) → θ)
pm5.21nd.3	⊢ (θ → (ψ ↔ χ))

Assertion

Ref	Expression
pm5.21nd	⊢ (φ → (ψ ↔ χ))

Proof of Theorem pm5.21nd

Step	Hyp	Ref	Expression
1		pm5.21nd.1	. . 3 ⊢ ((φ ∧ ψ) → θ)
2	1	ex 423	. 2 ⊢ (φ → (ψ → θ))
3		pm5.21nd.2	. . 3 ⊢ ((φ ∧ χ) → θ)
4	3	ex 423	. 2 ⊢ (φ → (χ → θ))
5		pm5.21nd.3	. . 3 ⊢ (θ → (ψ ↔ χ))
6	5	a1i 10	. 2 ⊢ (φ → (θ → (ψ ↔ χ)))
7	2, 4, 6	pm5.21ndd 343	1 ⊢ (φ → (ψ ↔ χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  ideqg  4869  ideqg2  4870  fvelimab  5371