cbval2

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbval2v if possible. (Contributed by NM, 22-Dec-2003) (Revised by Mario Carneiro, 6-Oct-2016) (Proof shortened by Wolf Lammen, 11-Sep-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbval2.1	⊢ Ⅎ 𝑧 𝜑
	cbval2.2	⊢ Ⅎ 𝑤 𝜑
	cbval2.3	⊢ Ⅎ 𝑥 𝜓
	cbval2.4	⊢ Ⅎ 𝑦 𝜓
	cbval2.5	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbval2	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbval2.1	⊢ Ⅎ 𝑧 𝜑
2		cbval2.2	⊢ Ⅎ 𝑤 𝜑
3		cbval2.3	⊢ Ⅎ 𝑥 𝜓
4		cbval2.4	⊢ Ⅎ 𝑦 𝜓
5		cbval2.5	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
6	1	nfal	⊢ Ⅎ 𝑧 ∀ 𝑦 𝜑
7	3	nfal	⊢ Ⅎ 𝑥 ∀ 𝑤 𝜓
8		nfv	⊢ Ⅎ 𝑦 𝑥 = 𝑧
9		nfv	⊢ Ⅎ 𝑤 𝑥 = 𝑧
10	2	a1i	⊢ ( 𝑥 = 𝑧 → Ⅎ 𝑤 𝜑 )
11	4	a1i	⊢ ( 𝑥 = 𝑧 → Ⅎ 𝑦 𝜓 )
12	5	ex	⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝑤 → ( 𝜑 ↔ 𝜓 ) ) )
13	8 9 10 11 12	cbv2	⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑦 𝜑 ↔ ∀ 𝑤 𝜓 ) )
14	6 7 13	cbval	⊢ ( ∀ 𝑥 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 𝜓 )

Metamath Proof Explorer

Theorem cbval2

Proof