Theorem sbhypf

Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004) (Proof shortened by Wolf Lammen, 25-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		sbhypf.1	$$⊢Ⅎx\psi$$
2		sbhypf.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3	2	sbimi	$$⊢\left[y/x\right]x=A\to \left[y/x\right]\left(\phi \leftrightarrow \psi \right)$$
4		eqsb1	$$⊢\left[y/x\right]x=A\leftrightarrow y=A$$
5	1	sbf	$$⊢\left[y/x\right]\psi \leftrightarrow \psi$$
6	5	sblbis	$$⊢\left[y/x\right]\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$
7	3 4 6	3imtr3i	$$⊢y=A\to \left(\left[y/x\right]\phi \leftrightarrow \psi \right)$$