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Metamath Proof Explorer

Theorem spimvALT

Description: Alternate proof of spimv . Note that it requires only ax-1 through ax-5 together with ax6e . Currently, proofs derive from ax6v , but if ax-6 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993) Remove dependency on ax-10 . (Revised by BJ, 29-Nov-2020) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	spimv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
	Assertion	spimvALT	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		spimv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
2		ax6e	⊢ ∃ 𝑥 𝑥 = 𝑦
3	2 1	eximii	⊢ ∃ 𝑥 ( 𝜑 → 𝜓 )
4	3	19.36iv	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )