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	⊢ Ⅎ 𝑥 𝜓
	equsal.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsexALT	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	⊢ Ⅎ 𝑥 𝜓
2		equsal.2	⊢ ( 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		ax6e	⊢ ∃ 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 )