Theorem stdpc4ALT

Description: Alternate proof of stdpc4 , shorter but using additional axioms. (Contributed by WL, 5-Jun-2026) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	stdpc4ALT	$$⊢\forall x\phi \to \left[t/x\right]\phi$$

Proof

Step	Hyp	Ref	Expression
1		ala1	$$⊢\forall x\phi \to \forall x\left(x=y\to \phi \right)$$
2	1	a1d	$$⊢\forall x\phi \to \left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
3	2	alrimiv	$$⊢\forall x\phi \to \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
4		dfsb	$$⊢\left[t/x\right]\phi \leftrightarrow \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
5	3 4	sylibr	$$⊢\forall x\phi \to \left[t/x\right]\phi$$