Theorem pm11.57

Description: Theorem *11.57 in WhiteheadRussell p. 165. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm11.57	$$⊢\forall x\phi \leftrightarrow \forall x\forall y\left(\phi \wedge \left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎy\phi$$
2	1	nfal	$$⊢Ⅎy\forall x\phi$$
3		sp	$$⊢\forall x\phi \to \phi$$
4		stdpc4	$$⊢\forall x\phi \to \left[y/x\right]\phi$$
5	3 4	jca	$$⊢\forall x\phi \to \left(\phi \wedge \left[y/x\right]\phi \right)$$
6	2 5	alrimi	$$⊢\forall x\phi \to \forall y\left(\phi \wedge \left[y/x\right]\phi \right)$$
7	6	axc4i	$$⊢\forall x\phi \to \forall x\forall y\left(\phi \wedge \left[y/x\right]\phi \right)$$
8		simpl	$$⊢\left(\phi \wedge \left[y/x\right]\phi \right)\to \phi$$
9	8	sps	$$⊢\forall y\left(\phi \wedge \left[y/x\right]\phi \right)\to \phi$$
10	9	alimi	$$⊢\forall x\forall y\left(\phi \wedge \left[y/x\right]\phi \right)\to \forall x\phi$$
11	7 10	impbii	$$⊢\forall x\phi \leftrightarrow \forall x\forall y\left(\phi \wedge \left[y/x\right]\phi \right)$$