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Theorem r19.30 2756
 Description: Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
Assertion
Ref Expression
r19.30 (x A (φ ψ) → (x A φ x A ψ))

Proof of Theorem r19.30
StepHypRef Expression
1 ralim 2685 . 2 (x Aψφ) → (x A ¬ ψx A φ))
2 orcom 376 . . . 4 ((φ ψ) ↔ (ψ φ))
3 df-or 359 . . . 4 ((ψ φ) ↔ (¬ ψφ))
42, 3bitri 240 . . 3 ((φ ψ) ↔ (¬ ψφ))
54ralbii 2638 . 2 (x A (φ ψ) ↔ x Aψφ))
6 orcom 376 . . 3 ((x A φ ¬ x A ¬ ψ) ↔ (¬ x A ¬ ψ x A φ))
7 dfrex2 2627 . . . 4 (x A ψ ↔ ¬ x A ¬ ψ)
87orbi2i 505 . . 3 ((x A φ x A ψ) ↔ (x A φ ¬ x A ¬ ψ))
9 imor 401 . . 3 ((x A ¬ ψx A φ) ↔ (¬ x A ¬ ψ x A φ))
106, 8, 93bitr4i 268 . 2 ((x A φ x A ψ) ↔ (x A ¬ ψx A φ))
111, 5, 103imtr4i 257 1 (x A (φ ψ) → (x A φ x A ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357  ∀wral 2614  ∃wrex 2615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by: (None)
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