Theorem equsexALT

Description: Alternate proof of equsex . This proves the result directly, instead of as a corollary of equsal via equs4 . Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 is ax6e . This proof mimics that of equsal (in particular, note that pm5.32i , exbii , 19.41 , mpbiran correspond respectively to pm5.74i , albii , 19.23 , a1bi ). (Contributed by BJ, 20-Aug-2020) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsal.1	$$⊢Ⅎx\psi$$
	equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsexALT	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsal.1	$$⊢Ⅎx\psi$$
2		equsal.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	pm5.32i	$$⊢\left(x=y\wedge \phi \right)\leftrightarrow \left(x=y\wedge \psi \right)$$
4	3	exbii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=y\wedge \psi \right)$$
5		ax6e	$$⊢\exists xx=y$$
6	1	19.41	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \left(\exists xx=y\wedge \psi \right)$$
7	5 6	mpbiran	$$⊢\exists x\left(x=y\wedge \psi \right)\leftrightarrow \psi$$
8	4 7	bitri	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$