Theorem bianfd 554

Description: A wff conjoined with falsehood is false.

Hypothesis

Ref	Expression
bianfd.1	⊢ (φ → ¬ ψ)

Assertion

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

Proof of Theorem bianfd

Step	Hyp	Ref	Expression
1		bianfd.1	. . 3 ⊢ (φ → ¬ ψ)
2	1	pm2.21d 74	. 2 ⊢ (φ → (ψ → (ψ ∧ χ)))
3		pm3.26 256	. . 3 ⊢ ((ψ ∧ χ) → ψ)
4	3	a1i 7	. 2 ⊢ (φ → ((ψ ∧ χ) → ψ))
5	2, 4	impbid 397	1 ⊢ (φ → (ψ ↔ (ψ ∧ χ)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196

This theorem is referenced by:  eueq2 1429  eueq3 1430

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

metamath.org