Theorem bimsc1 557

Description: Removal of conjunct from one side of an equivalence.

Assertion

Ref	Expression
bimsc1	⊢ (((φ → ψ) ∧ (χ ↔ (ψ ∧ φ))) → (χ ↔ φ))

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		id 9	. 2 ⊢ ((χ ↔ (ψ ∧ φ)) → (χ ↔ (ψ ∧ φ)))
2		pm4.71r 482	. . . 4 ⊢ ((φ → ψ) ↔ (φ ↔ (ψ ∧ φ)))
3	2	biimp 133	. . 3 ⊢ ((φ → ψ) → (φ ↔ (ψ ∧ φ)))
4	3	bicomd 399	. 2 ⊢ ((φ → ψ) → ((ψ ∧ φ) ↔ φ))
5	1, 4	sylan9bbr 419	1 ⊢ (((φ → ψ) ∧ (χ ↔ (ψ ∧ φ))) → (χ ↔ φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  bm1.3ii 1481

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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