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Metamath Proof Explorer

Theorem spv

Description: Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker spvv if possible. (Contributed by NM, 30-Aug-1993) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		spv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2	1	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
3	2	spimv	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )