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Metamath Proof Explorer

Theorem sbcbi

Description: Implication form of sbcbii . sbcbi is sbcbiVD without virtual deductions and was automatically derived from sbcbiVD using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	sbcbi	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		spsbc	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → [ 𝐴 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ) )
2		sbcbig	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] ( 𝜑 ↔ 𝜓 ) ↔ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )
3	1 2	sylibd	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝜓 ) → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜓 ) ) )