Theorem sbcom

Description: A commutativity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out sbcom3vv for a version requiring fewer axioms. (Contributed by NM, 27-May-1997) (Proof shortened by Wolf Lammen, 20-Sep-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	sbcom	$$⊢\left[y/z\right]\left[y/x\right]\phi \leftrightarrow \left[y/x\right]\left[y/z\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		sbco3	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/x\right]\left[x/z\right]\phi$$
2		sbcom3	$$⊢\left[y/z\right]\left[z/x\right]\phi \leftrightarrow \left[y/z\right]\left[y/x\right]\phi$$
3		sbcom3	$$⊢\left[y/x\right]\left[x/z\right]\phi \leftrightarrow \left[y/x\right]\left[y/z\right]\phi$$
4	1 2 3	3bitr3i	$$⊢\left[y/z\right]\left[y/x\right]\phi \leftrightarrow \left[y/x\right]\left[y/z\right]\phi$$