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Metamath Proof Explorer

Theorem cbvmo

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 cbvmow , cbvmovw when possible. (Contributed by NM, 9-Mar-1995) (Revised by Andrew Salmon, 8-Jun-2011) (Proof shortened by Wolf Lammen, 4-Jan-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvmo.1	⊢ Ⅎ 𝑦 𝜑
	cbvmo.2	⊢ Ⅎ 𝑥 𝜓
	cbvmo.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvmo	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvmo.1	⊢ Ⅎ 𝑦 𝜑
2		cbvmo.2	⊢ Ⅎ 𝑥 𝜓
3		cbvmo.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
4	1	sb8mo	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5	2 3	sbie	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 )
6	5	mobii	⊢ ( ∃* 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃* 𝑦 𝜓 )
7	4 6	bitri	⊢ ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )