Theorem cbvalw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017)

	Ref	Expression
Hypotheses	cbvalw.1	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
	cbvalw.2	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
	cbvalw.3	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
	cbvalw.4	⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 )
	cbvalw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvalw	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvalw.1	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
2		cbvalw.2	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
3		cbvalw.3	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 ∀ 𝑦 𝜓 )
4		cbvalw.4	⊢ ( ¬ 𝜑 → ∀ 𝑦 ¬ 𝜑 )
5		cbvalw.5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
6	5	biimpd	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
7	1 2 6	cbvaliw	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )
8	5	biimprd	⊢ ( 𝑥 = 𝑦 → ( 𝜓 → 𝜑 ) )
9	8	equcoms	⊢ ( 𝑦 = 𝑥 → ( 𝜓 → 𝜑 ) )
10	3 4 9	cbvaliw	⊢ ( ∀ 𝑦 𝜓 → ∀ 𝑥 𝜑 )
11	7 10	impbii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 𝜓 )