Metamath Proof Explorer

Theorem ancomstVD

Description: Closed form of ancoms . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	\|- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
qed:1,?: e0a	\|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ps /\ ph ) -> ch ) )

The proof of ancomst is derived automatically from it. (Contributed by Alan Sare, 25-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	ancomstVD	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ancom	⊢ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) )
2		imbi1	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜓 ∧ 𝜑 ) ) → ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) ) )
3	1 2	e0a	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → 𝜒 ) ↔ ( ( 𝜓 ∧ 𝜑 ) → 𝜒 ) )