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	$$⊢A\in V\to \left(\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		spsbc	$$⊢A\in V\to \left(\forall x\left(\phi \leftrightarrow \psi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\right)$$
2		sbcbig	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
3	1 2	sylibd	$$⊢A\in V\to \left(\forall x\left(\phi \leftrightarrow \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$