Theorem pm13.196a

Description: Theorem *13.196 in WhiteheadRussell p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011)

	Ref	Expression
Assertion	pm13.196a	$$⊢\neg \phi \leftrightarrow \forall y\left(\left[y/x\right]\phi \to y\ne x\right)$$

Proof

Step	Hyp	Ref	Expression
1		sbelx	$$⊢\neg \phi \leftrightarrow \exists y\left(y=x\wedge \left[y/x\right]\neg \phi \right)$$
2		sbalex	$$⊢\exists y\left(y=x\wedge \left[y/x\right]\neg \phi \right)\leftrightarrow \forall y\left(y=x\to \left[y/x\right]\neg \phi \right)$$
3		sbn	$$⊢\left[y/x\right]\neg \phi \leftrightarrow \neg \left[y/x\right]\phi$$
4	3	imbi2i	$$⊢\left(y=x\to \left[y/x\right]\neg \phi \right)\leftrightarrow \left(y=x\to \neg \left[y/x\right]\phi \right)$$
5		con2b	$$⊢\left(y=x\to \neg \left[y/x\right]\phi \right)\leftrightarrow \left(\left[y/x\right]\phi \to \neg y=x\right)$$
6		df-ne	$$⊢y\ne x\leftrightarrow \neg y=x$$
7	6	bicomi	$$⊢\neg y=x\leftrightarrow y\ne x$$
8	7	imbi2i	$$⊢\left(\left[y/x\right]\phi \to \neg y=x\right)\leftrightarrow \left(\left[y/x\right]\phi \to y\ne x\right)$$
9	4 5 8	3bitri	$$⊢\left(y=x\to \left[y/x\right]\neg \phi \right)\leftrightarrow \left(\left[y/x\right]\phi \to y\ne x\right)$$
10	9	albii	$$⊢\forall y\left(y=x\to \left[y/x\right]\neg \phi \right)\leftrightarrow \forall y\left(\left[y/x\right]\phi \to y\ne x\right)$$
11	1 2 10	3bitri	$$⊢\neg \phi \leftrightarrow \forall y\left(\left[y/x\right]\phi \to y\ne x\right)$$