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Theorem rexim 2718
 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 con3 126 . . . 4 ((φψ) → (¬ ψ → ¬ φ))
21ral2imi 2690 . . 3 (x A (φψ) → (x A ¬ ψx A ¬ φ))
32con3d 125 . 2 (x A (φψ) → (¬ x A ¬ φ → ¬ x A ¬ ψ))
4 dfrex2 2627 . 2 (x A φ ↔ ¬ x A ¬ φ)
5 dfrex2 2627 . 2 (x A ψ ↔ ¬ x A ¬ ψ)
63, 4, 53imtr4g 261 1 (x A (φψ) → (x A φx A ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2619  df-rex 2620 This theorem is referenced by:  reximia  2719  reximdai  2722  r19.29  2754  reupick2  3541  ss2iun  3984
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