Theorem bj-ssblem2

Description: An instance of ax-11 proved without it. The converse may not be provable without ax-11 (since using alcomimw would require a DV on ph , x , which defeats the purpose). (Contributed by BJ, 22-Dec-2020) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ssblem2	$$⊢\forall x\forall y\left(y=t\to \left(x=y\to \phi \right)\right)\to \forall y\forall x\left(y=t\to \left(x=y\to \phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		equequ1	$$⊢y=z\to \left(y=t\leftrightarrow z=t\right)$$
2		equequ2	$$⊢y=z\to \left(x=y\leftrightarrow x=z\right)$$
3	2	imbi1d	$$⊢y=z\to \left(\left(x=y\to \phi \right)\leftrightarrow \left(x=z\to \phi \right)\right)$$
4	1 3	imbi12d	$$⊢y=z\to \left(\left(y=t\to \left(x=y\to \phi \right)\right)\leftrightarrow \left(z=t\to \left(x=z\to \phi \right)\right)\right)$$
5	4	alcomimw	$$⊢\forall x\forall y\left(y=t\to \left(x=y\to \phi \right)\right)\to \forall y\forall x\left(y=t\to \left(x=y\to \phi \right)\right)$$