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Metamath Proof Explorer

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	$⊢ \neg φ \leftrightarrow \forall y ([y/ x] φ \to y \neq x)$

Proof

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