Theorem bj-equsal1t

Description: Duplication of wl-equsal1t , with shorter proof. If one imposes a disjoint variable condition on x , y , then one can use alequexv and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom is also interesting. (Contributed by BJ, 6-Oct-2018)

	Ref	Expression
Assertion	bj-equsal1t	$$⊢Ⅎx\phi \to \left(\forall x\left(x=y\to \phi \right)\leftrightarrow \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-alequex	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\phi$$
2		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
3	1 2	imbitrid	$$⊢Ⅎx\phi \to \left(\forall x\left(x=y\to \phi \right)\to \phi \right)$$
4		nf5r	$$⊢Ⅎx\phi \to \left(\phi \to \forall x\phi \right)$$
5		ala1	$$⊢\forall x\phi \to \forall x\left(x=y\to \phi \right)$$
6	4 5	syl6	$$⊢Ⅎx\phi \to \left(\phi \to \forall x\left(x=y\to \phi \right)\right)$$
7	3 6	impbid	$$⊢Ⅎx\phi \to \left(\forall x\left(x=y\to \phi \right)\leftrightarrow \phi \right)$$