ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ralim GIF version

Theorem ralim 2374
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
Assertion
Ref Expression
ralim (x A (φψ) → (x A φx A ψ))

Proof of Theorem ralim
StepHypRef Expression
1 df-ral 2305 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
2 ax-2 6 . . . 4 ((x A → (φψ)) → ((x Aφ) → (x Aψ)))
32al2imi 1344 . . 3 (x(x A → (φψ)) → (x(x Aφ) → x(x Aψ)))
41, 3sylbi 114 . 2 (x A (φψ) → (x(x Aφ) → x(x Aψ)))
5 df-ral 2305 . 2 (x A φx(x Aφ))
6 df-ral 2305 . 2 (x A ψx(x Aψ))
74, 5, 63imtr4g 194 1 (x A (φψ) → (x A φx A ψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   wcel 1390  wral 2300
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
This theorem depends on definitions:  df-bi 110  df-ral 2305
This theorem is referenced by:  ral2imi  2379  trint  3860  peano2  4261  mpteqb  5204  lbzbi  8327
  Copyright terms: Public domain W3C validator