Metamath Proof Explorer

Theorem bj-a1k

Description: Weakening of ax-1 . As a consequence, its associated inference is an instance (where we allow extra hypotheses) of ax-1 . Its commuted form is 2a1 (but bj-a1k does not require ax-2 ). This shortens the proofs of dfwe2 (937>925), ordunisuc2 (789>777), r111 (558>545), smo11 (1176>1164). (Contributed by BJ, 11-Aug-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	bj-a1k	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-1	⊢ ( 𝜓 → ( 𝜒 → 𝜓 ) )
2	1	a1i	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → 𝜓 ) ) )