Theorem bj-cbvex

Description: Changing a bound variable (existential quantification case) in a weak axiomatization that assumes that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023) Proved from ax-1 -- ax-5 . (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-cbval.denote	⊢ ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦
	bj-cbval.denote2	⊢ ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥
	bj-cbval.equcomiv	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
	bj-cbval.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	bj-cbval.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
	bj-cbval.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	bj-cbvex	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-cbval.denote	⊢ ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦
2		bj-cbval.denote2	⊢ ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥
3		bj-cbval.equcomiv	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
4		bj-cbval.nf0	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
5		bj-cbval.nf1	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
6		bj-cbval.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
7		ax5d	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑦 𝜓 ) )
8	2	a1i	⊢ ( 𝜑 → ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥 )
9	3 6	sylan2	⊢ ( ( 𝜑 ∧ 𝑦 = 𝑥 ) → ( 𝜓 ↔ 𝜒 ) )
10	9	biimpd	⊢ ( ( 𝜑 ∧ 𝑦 = 𝑥 ) → ( 𝜓 → 𝜒 ) )
11	4 5 7 8 10	bj-cbveximdv	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑦 𝜒 ) )
12		ax5d	⊢ ( 𝜑 → ( 𝜒 → ∀ 𝑥 𝜒 ) )
13	1	a1i	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦 )
14	6	biimprd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜒 → 𝜓 ) )
15	5 4 12 13 14	bj-cbveximdv	⊢ ( 𝜑 → ( ∃ 𝑦 𝜒 → ∃ 𝑥 𝜓 ) )
16	11 15	impbid	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 ↔ ∃ 𝑦 𝜒 ) )