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Theorem r19.41 2763
 Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
Hypothesis
Ref Expression
r19.41.1 xψ
Assertion
Ref Expression
r19.41 (x A (φ ψ) ↔ (x A φ ψ))

Proof of Theorem r19.41
StepHypRef Expression
1 anass 630 . . . 4 (((x A φ) ψ) ↔ (x A (φ ψ)))
21exbii 1582 . . 3 (x((x A φ) ψ) ↔ x(x A (φ ψ)))
3 r19.41.1 . . . 4 xψ
4319.41 1879 . . 3 (x((x A φ) ψ) ↔ (x(x A φ) ψ))
52, 4bitr3i 242 . 2 (x(x A (φ ψ)) ↔ (x(x A φ) ψ))
6 df-rex 2620 . 2 (x A (φ ψ) ↔ x(x A (φ ψ)))
7 df-rex 2620 . . 3 (x A φx(x A φ))
87anbi1i 676 . 2 ((x A φ ψ) ↔ (x(x A φ) ψ))
95, 6, 83bitr4i 268 1 (x A (φ ψ) ↔ (x A φ ψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃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-6 1729  ax-11 1746 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2620 This theorem is referenced by:  r19.41v  2764
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