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Theorem rexim 2407
Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
rexim (x A (φψ) → (x A φx A ψ))

Proof of Theorem rexim
StepHypRef Expression
1 df-ral 2305 . . . 4 (x A (φψ) ↔ x(x A → (φψ)))
2 simpl 102 . . . . . . 7 ((x A φ) → x A)
32a1i 9 . . . . . 6 ((x A → (φψ)) → ((x A φ) → x A))
4 pm3.31 249 . . . . . 6 ((x A → (φψ)) → ((x A φ) → ψ))
53, 4jcad 291 . . . . 5 ((x A → (φψ)) → ((x A φ) → (x A ψ)))
65alimi 1341 . . . 4 (x(x A → (φψ)) → x((x A φ) → (x A ψ)))
71, 6sylbi 114 . . 3 (x A (φψ) → x((x A φ) → (x A ψ)))
8 exim 1487 . . 3 (x((x A φ) → (x A ψ)) → (x(x A φ) → x(x A ψ)))
97, 8syl 14 . 2 (x A (φψ) → (x(x A φ) → x(x A ψ)))
10 df-rex 2306 . 2 (x A φx(x A φ))
11 df-rex 2306 . 2 (x A ψx(x A ψ))
129, 10, 113imtr4g 194 1 (x A (φψ) → (x A φx A ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240  wex 1378   wcel 1390  wral 2300  wrex 2301
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-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-ral 2305  df-rex 2306
This theorem is referenced by:  reximia  2408  reximdai  2411  r19.29  2444  reupick2  3217  ss2iun  3663  chfnrn  5221
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