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Theorem r19.35 2758
 Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
r19.35 (x A (φψ) ↔ (x A φx A ψ))

Proof of Theorem r19.35
StepHypRef Expression
1 r19.26 2746 . . . 4 (x A (φ ¬ ψ) ↔ (x A φ x A ¬ ψ))
2 annim 414 . . . . 5 ((φ ¬ ψ) ↔ ¬ (φψ))
32ralbii 2638 . . . 4 (x A (φ ¬ ψ) ↔ x A ¬ (φψ))
4 df-an 360 . . . 4 ((x A φ x A ¬ ψ) ↔ ¬ (x A φ → ¬ x A ¬ ψ))
51, 3, 43bitr3i 266 . . 3 (x A ¬ (φψ) ↔ ¬ (x A φ → ¬ x A ¬ ψ))
65con2bii 322 . 2 ((x A φ → ¬ x A ¬ ψ) ↔ ¬ x A ¬ (φψ))
7 dfrex2 2627 . . 3 (x A ψ ↔ ¬ x A ¬ ψ)
87imbi2i 303 . 2 ((x A φx A ψ) ↔ (x A φ → ¬ x A ¬ ψ))
9 dfrex2 2627 . 2 (x A (φψ) ↔ ¬ x A ¬ (φψ))
106, 8, 93bitr4ri 269 1 (x A (φψ) ↔ (x A φx A ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620 This theorem is referenced by:  r19.36av  2759  r19.37  2760  r19.43  2766  r19.37zv  3646  r19.36zv  3650
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