Theorem pm13.14

Description: Theorem *13.14 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011)

	Ref	Expression
Assertion	pm13.14	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \neg \phi \right)\to x\ne A$$

Proof

Step	Hyp	Ref	Expression
1		sbceq1a	$$⊢x=A\to \left(\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
2	1	biimprcd	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(x=A\to \phi \right)$$
3	2	necon3bd	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\neg \phi \to x\ne A\right)$$
4	3	imp	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \wedge \neg \phi \right)\to x\ne A$$