Theorem pm4.45im 267

Description: Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.

Assertion

Ref	Expression
pm4.45im	⊢ (φ ↔ (φ ∧ (ψ → φ)))

Proof of Theorem pm4.45im

Step	Hyp	Ref	Expression
1		ax-1 3	. . 3 ⊢ (φ → (ψ → φ))
2	1	ancli 244	. 2 ⊢ (φ → (φ ∧ (ψ → φ)))
3		pm3.26 256	. 2 ⊢ ((φ ∧ (ψ → φ)) → φ)
4	2, 3	impbi 139	1 ⊢ (φ ↔ (φ ∧ (ψ → φ)))

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

This theorem is referenced by:  difdif 1595

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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