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 𝑦 𝜑
cbvmo.2 𝑥 𝜓
cbvmo.3 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion cbvmo ( ∃* 𝑥 𝜑 ↔ ∃* 𝑦 𝜓 )

Proof

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