Theorem sbrim 1827

Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
sbrim.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem sbrim

Step	Hyp	Ref	Expression
1		sbim 1824	. 2 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ))
2		sbrim.1	. . . 4 ⊢ (φ → ∀xφ)
3	2	sbh 1656	. . 3 ⊢ ([y / x]φ ↔ φ)
4	3	imbi1i 227	. 2 ⊢ (([y / x]φ → [y / x]ψ) ↔ (φ → [y / x]ψ))
5	1, 4	bitri 173	1 ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbco2d  1837  sbco2vd  1838  hbsbd  1855