Theorem sbbiiev

Description: An equivalence of substitutions (as in sbbii ) allowing the additional information that x = t . Version of sbiev and sbievw without a disjoint variable condition on ps , useful for substituting only part of ph . (Contributed by SN, 24-Aug-2025)

	Ref	Expression
Hypothesis	sbbiiev.1	$$⊢x=t\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	sbbiiev	$$⊢\left[t/x\right]\phi \leftrightarrow \left[t/x\right]\psi$$

Proof

Step	Hyp	Ref	Expression
1		sbbiiev.1	$$⊢x=t\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.74i	$$⊢\left(x=t\to \phi \right)\leftrightarrow \left(x=t\to \psi \right)$$
3	2	albii	$$⊢\forall x\left(x=t\to \phi \right)\leftrightarrow \forall x\left(x=t\to \psi \right)$$
4		sb6	$$⊢\left[t/x\right]\phi \leftrightarrow \forall x\left(x=t\to \phi \right)$$
5		sb6	$$⊢\left[t/x\right]\psi \leftrightarrow \forall x\left(x=t\to \psi \right)$$
6	3 4 5	3bitr4i	$$⊢\left[t/x\right]\phi \leftrightarrow \left[t/x\right]\psi$$