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Metamath Proof Explorer

Theorem sb5rf

Description: Reversed substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 3-Feb-2005) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 20-Sep-2018) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb5rf.1	$⊢ Ⅎ y φ$
	Assertion	sb5rf	$⊢ φ \leftrightarrow \exists y (y = x \land [y/ x] φ)$

Proof

Step	Hyp	Ref	Expression
1		sb5rf.1	$⊢ Ⅎ y φ$
2		sbequ12r	$⊢ y = x \to ([y/ x] φ \leftrightarrow φ)$
3	1 2	equsex	$⊢ \exists y (y = x \land [y/ x] φ) \leftrightarrow φ$
4	3	bicomi	$⊢ φ \leftrightarrow \exists y (y = x \land [y/ x] φ)$