Theorem sb6rf

Description: Reversed substitution. For a version requiring disjoint variables, but fewer axioms, see sb6rfv . Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sb6rfv if possible. (Contributed by NM, 1-Aug-1993) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 21-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb5rf.1	$$⊢Ⅎy\phi$$
Assertion	sb6rf	$$⊢\phi \leftrightarrow \forall y\left(y=x\to \left[y/x\right]\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb5rf.1	$$⊢Ⅎy\phi$$
2		sbequ12r	$$⊢y=x\to \left(\left[y/x\right]\phi \leftrightarrow \phi \right)$$
3	1 2	equsal	$$⊢\forall y\left(y=x\to \left[y/x\right]\phi \right)\leftrightarrow \phi$$
4	3	bicomi	$$⊢\phi \leftrightarrow \forall y\left(y=x\to \left[y/x\right]\phi \right)$$