Theorem sblimv 1771

Description: Version of sblim 1828 where x and y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)

Hypothesis

Ref	Expression
sblimv.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
sblimv	⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → ψ))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem sblimv

Step	Hyp	Ref	Expression
1		sbimv 1770	. 2 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ))
2		sblimv.1	. . . 4 ⊢ (ψ → ∀xψ)
3	2	sbh 1656	. . 3 ⊢ ([y / x]ψ ↔ ψ)
4	3	imbi2i 215	. 2 ⊢ (([y / x]φ → [y / x]ψ) ↔ ([y / x]φ → ψ))
5	1, 4	bitri 173	1 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → ψ))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-sb 1643

This theorem is referenced by: (None)