Metamath Proof Explorer


Theorem cbvald

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim . Usage of this theorem is discouraged because it depends on ax-13 . See cbvaldw for a version with x , y disjoint, not depending on ax-13 . (Contributed by NM, 2-Jan-2002) (Revised by Mario Carneiro, 6-Oct-2016) (Revised by Wolf Lammen, 13-May-2018) (New usage is discouraged.)

Ref Expression
Hypotheses cbvald.1 𝑦 𝜑
cbvald.2 ( 𝜑 → Ⅎ 𝑦 𝜓 )
cbvald.3 ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓𝜒 ) ) )
Assertion cbvald ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )

Proof

Step Hyp Ref Expression
1 cbvald.1 𝑦 𝜑
2 cbvald.2 ( 𝜑 → Ⅎ 𝑦 𝜓 )
3 cbvald.3 ( 𝜑 → ( 𝑥 = 𝑦 → ( 𝜓𝜒 ) ) )
4 nfv 𝑥 𝜑
5 nfvd ( 𝜑 → Ⅎ 𝑥 𝜒 )
6 4 1 2 5 3 cbv2 ( 𝜑 → ( ∀ 𝑥 𝜓 ↔ ∀ 𝑦 𝜒 ) )