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Theorem r19.43 2766
 Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
Assertion
Ref Expression
r19.43 (x A (φ ψ) ↔ (x A φ x A ψ))

Proof of Theorem r19.43
StepHypRef Expression
1 r19.35 2758 . 2 (x Aφψ) ↔ (x A ¬ φx A ψ))
2 df-or 359 . . 3 ((φ ψ) ↔ (¬ φψ))
32rexbii 2639 . 2 (x A (φ ψ) ↔ x Aφψ))
4 df-or 359 . . 3 ((x A φ x A ψ) ↔ (¬ x A φx A ψ))
5 ralnex 2624 . . . 4 (x A ¬ φ ↔ ¬ x A φ)
65imbi1i 315 . . 3 ((x A ¬ φx A ψ) ↔ (¬ x A φx A ψ))
74, 6bitr4i 243 . 2 ((x A φ x A ψ) ↔ (x A ¬ φx A ψ))
81, 3, 73bitr4i 268 1 (x A (φ ψ) ↔ (x A φ x A ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ 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:  r19.44av  2767  r19.45av  2768  r19.45zv  3647  iunun  4046  nncdiv3  6277
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