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

Theorem ralbida 2314
Description: Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
Hypotheses
Ref Expression
ralbida.1 xφ
ralbida.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
ralbida (φ → (x A ψx A χ))

Proof of Theorem ralbida
StepHypRef Expression
1 ralbida.1 . . 3 xφ
2 ralbida.2 . . . 4 ((φ x A) → (ψχ))
32pm5.74da 417 . . 3 (φ → ((x Aψ) ↔ (x Aχ)))
41, 3albid 1503 . 2 (φ → (x(x Aψ) ↔ x(x Aχ)))
5 df-ral 2305 . 2 (x A ψx(x Aψ))
6 df-ral 2305 . 2 (x A χx(x Aχ))
74, 5, 63bitr4g 212 1 (φ → (x A ψx A χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240  wnf 1346   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  ax-4 1397
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305
This theorem is referenced by:  ralbidva  2316  ralbid  2318  2ralbida  2339  ralbi  2439
  Copyright terms: Public domain W3C validator