Theorem bj-cbval

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

	Ref	Expression
Hypotheses	bj-cbval.denote	⊢ ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦
	bj-cbval.denote2	⊢ ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥
	bj-cbval.maj	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	bj-cbval.equcomiv	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
Assertion	bj-cbval	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-cbval.denote	⊢ ∀ 𝑦 ∃ 𝑥 𝑥 = 𝑦
2		bj-cbval.denote2	⊢ ∀ 𝑥 ∃ 𝑦 𝑦 = 𝑥
3		bj-cbval.maj	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4		bj-cbval.equcomiv	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
5	3	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
6	5 1	bj-cbvalimi	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )
7	3	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
8	4 7	syl	⊢ ( 𝑦 = 𝑥 → ( 𝜓 → 𝜑 ) )
9	8 2	bj-cbvalimi	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜑 )
10	6 9	impbii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )