Theorem sbiedvw

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbievw ). Version of sbied and sbiedv with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Gino Giotto, 29-Jan-2024)

	Ref	Expression
Hypothesis	sbiedvw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	sbiedvw	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbiedvw.1	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
2		sbrimvw	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \left[y/x\right]\psi \right)$$
3	1	expcom	$$⊢x=y\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$
4	3	pm5.74d	$$⊢x=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
5	4	sbievw	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)$$
6	2 5	bitr3i	$$⊢\left(\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\phi \to \chi \right)$$
7	6	pm5.74ri	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$