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	$$⊢Ⅎy\phi$$
Assertion	sb5f	$$⊢\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb6f.1	$$⊢Ⅎy\phi$$
2	1	sb6f	$$⊢\left[y/x\right]\phi \leftrightarrow \forall x\left(x=y\to \phi \right)$$
3	1	equs45f	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \forall x\left(x=y\to \phi \right)$$
4	2 3	bitr4i	$$⊢\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$