Theorem pm10.52

Description: Theorem *10.52 in WhiteheadRussell p. 155. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.52	$$⊢\exists x\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		19.23v	$$⊢\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)$$
2		pm5.5	$$⊢\exists x\phi \to \left(\left(\exists x\phi \to \psi \right)\leftrightarrow \psi \right)$$
3	1 2	syl5bb	$$⊢\exists x\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \psi \right)$$