Theorem bj-cbval

Description: Changing a bound variable (universal 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-cbval	⊢ ( 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		ax5e	⊢ ( ∃ 𝑥 𝜒 → 𝜒 )
8	7	a1i	⊢ ( 𝜑 → ( ∃ 𝑥 𝜒 → 𝜒 ) )
9	1	a1i	⊢ ( 𝜑 → ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦 )
10	6	biimpd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 → 𝜒 ) )
11	4 5 8 9 10	bj-cbvalimdv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑦 𝜒 ) )
12		ax5e	⊢ ( ∃ 𝑦 𝜓 → 𝜓 )
13	12	a1i	⊢ ( 𝜑 → ( ∃ 𝑦 𝜓 → 𝜓 ) )
14	2	a1i	⊢ ( 𝜑 → ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥 )
15	6	biimprd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜒 → 𝜓 ) )
16	3 15	sylan2	⊢ ( ( 𝜑 ∧ 𝑦 = 𝑥 ) → ( 𝜒 → 𝜓 ) )
17	5 4 13 14 16	bj-cbvalimdv	⊢ ( 𝜑 → ( ∀ 𝑦 𝜒 → ∀ 𝑥 𝜓 ) )
18	11 17	impbid	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )