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Metamath Proof Explorer

Theorem spsbcd

Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of Quine p. 44. See also stdpc4 and rspsbc . (Contributed by Mario Carneiro, 9-Feb-2017)

	Ref	Expression
Hypotheses	spsbcd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	spsbcd.2	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
Assertion	spsbcd	⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		spsbcd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		spsbcd.2	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
3		spsbc	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 𝜓 → [ 𝐴 / 𝑥 ] 𝜓 ) )
4	1 2 3	sylc	⊢ ( 𝜑 → [ 𝐴 / 𝑥 ] 𝜓 )