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	$⊢ Ⅎ x ψ$
	equsal.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	equsexALT	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$

Proof

Step	Hyp	Ref	Expression
1		equsal.1	$⊢ Ⅎ x ψ$
2		equsal.2	$⊢ x = y \to (φ \leftrightarrow ψ)$
3	2	pm5.32i	$⊢ (x = y \land φ) \leftrightarrow (x = y \land ψ)$
4	3	exbii	$⊢ \exists x (x = y \land φ) \leftrightarrow \exists x (x = y \land ψ)$
5		ax6e	$⊢ \exists x x = y$
6	1	19.41	$⊢ \exists x (x = y \land ψ) \leftrightarrow (\exists x x = y \land ψ)$
7	5 6	mpbiran	$⊢ \exists x (x = y \land ψ) \leftrightarrow ψ$
8	4 7	bitri	$⊢ \exists x (x = y \land φ) \leftrightarrow ψ$

Metamath Proof Explorer

Theorem equsexALT

Proof