Theorem sb7f

Description: This version of dfsb7 does not require that ph and z be disjoint. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 , i.e., that does not have the concept of a variable not occurring in a formula. (Definition dfsb1 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 26-Jul-2006) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	sb7f.1	$$⊢Ⅎz\phi$$
Assertion	sb7f	$$⊢\left[y/x\right]\phi \leftrightarrow \exists z\left(z=y\wedge \exists x\left(x=z\wedge \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		sb7f.1	$$⊢Ⅎz\phi$$
2	1	sb5f	$$⊢\left[z/x\right]\phi \leftrightarrow \exists x\left(x=z\wedge \phi \right)$$
3	2	sbbii	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/z\right]\exists x\left(x=z\wedge \phi \right)$$
4	1	sbco2	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\phi$$
5		sb5	$$⊢\left[y/z\right]\exists x\left(x=z\wedge \phi \right)\leftrightarrow \exists z\left(z=y\wedge \exists x\left(x=z\wedge \phi \right)\right)$$
6	3 4 5	3bitr3i	$$⊢\left[y/x\right]\phi \leftrightarrow \exists z\left(z=y\wedge \exists x\left(x=z\wedge \phi \right)\right)$$