Theorem equsexv

Description: An equivalence related to implicit substitution. Version of equsex with a disjoint variable condition, which does not require ax-13 . See equsexvw for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv . (Contributed by NM, 5-Aug-1993) (Revised by BJ, 31-May-2019)

	Ref	Expression
Hypotheses	equsalv.nf	⊢ Ⅎ 𝑥 𝜓
	equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsalv.nf	⊢ Ⅎ 𝑥 𝜓
2		equsalv.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	2	pm5.32i	⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ( 𝑥 = 𝑦 ∧ 𝜓 ) )
4	3	exbii	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) )
5		ax6ev	⊢ ∃ 𝑥 𝑥 = 𝑦
6	1	19.41	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝑦 ∧ 𝜓 ) )
7	5 6	mpbiran	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜓 ) ↔ 𝜓 )
8	4 7	bitri	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )