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Metamath Proof Explorer

Theorem spimv

Description: A version of spim with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv and spimvw for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993) Usage of this theorem is discouraged because it depends on ax-13 . Use spimvw instead. (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spimv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	2 1	spim	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )