Metamath Proof Explorer


Theorem cbvralsvwOLD

Description: Obsolete version of cbvralsvw as of 8-Mar-2025. (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion cbvralsvwOLD ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 nfv 𝑧 𝜑
2 nfs1v 𝑥 [ 𝑧 / 𝑥 ] 𝜑
3 sbequ12 ( 𝑥 = 𝑧 → ( 𝜑 ↔ [ 𝑧 / 𝑥 ] 𝜑 ) )
4 1 2 3 cbvralw ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑧𝐴 [ 𝑧 / 𝑥 ] 𝜑 )
5 nfv 𝑦 [ 𝑧 / 𝑥 ] 𝜑
6 nfv 𝑧 [ 𝑦 / 𝑥 ] 𝜑
7 sbequ ( 𝑧 = 𝑦 → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
8 5 6 7 cbvralw ( ∀ 𝑧𝐴 [ 𝑧 / 𝑥 ] 𝜑 ↔ ∀ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )
9 4 8 bitri ( ∀ 𝑥𝐴 𝜑 ↔ ∀ 𝑦𝐴 [ 𝑦 / 𝑥 ] 𝜑 )