This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sb5f

Description: Equivalence for substitution when y is not free in ph . The implication "to the right" is sb1 and does not require the nonfreeness hypothesis. Theorem sb5 replaces the nonfreeness hypothesis with a disjoint variable condition on x , y and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 4-Oct-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sb6f.1	$⊢ Ⅎ y φ$
	Assertion	sb5f	$⊢ [y/ x] φ \leftrightarrow \exists x (x = y \land φ)$

Proof

Step	Hyp	Ref	Expression
1		sb6f.1	$⊢ Ⅎ y φ$
2	1	sb6f	$⊢ [y/ x] φ \leftrightarrow \forall x (x = y \to φ)$
3	1	equs45f	$⊢ \exists x (x = y \land φ) \leftrightarrow \forall x (x = y \to φ)$
4	2 3	bitr4i	$⊢ [y/ x] φ \leftrightarrow \exists x (x = y \land φ)$