Theorem sbied

Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie ) Usage of this theorem is discouraged because it depends on ax-13 . See sbiedw , sbiedvw for variants using disjoint variables, but requiring fewer axioms. (Contributed by NM, 30-Jun-1994) (Revised by Mario Carneiro, 4-Oct-2016) (Proof shortened by Wolf Lammen, 24-Jun-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	sbied.1	$$⊢Ⅎx\phi$$
	sbied.2	$$⊢\phi \to Ⅎx\chi$$
	sbied.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
Assertion	sbied	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sbied.1	$$⊢Ⅎx\phi$$
2		sbied.2	$$⊢\phi \to Ⅎx\chi$$
3		sbied.3	$$⊢\phi \to \left(x=y\to \left(\psi \leftrightarrow \chi \right)\right)$$
4	1	sbrim	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \left[y/x\right]\psi \right)$$
5	1 2	nfim1	$$⊢Ⅎx\left(\phi \to \chi \right)$$
6	3	com12	$$⊢x=y\to \left(\phi \to \left(\psi \leftrightarrow \chi \right)\right)$$
7	6	pm5.74d	$$⊢x=y\to \left(\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)\right)$$
8	5 7	sbie	$$⊢\left[y/x\right]\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \chi \right)$$
9	4 8	bitr3i	$$⊢\left(\phi \to \left[y/x\right]\psi \right)\leftrightarrow \left(\phi \to \chi \right)$$
10	9	pm5.74ri	$$⊢\phi \to \left(\left[y/x\right]\psi \leftrightarrow \chi \right)$$