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Theorem r19.41 2439
 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 383 . . . 4 (((x A φ) ψ) ↔ (x A (φ ψ)))
21exbii 1474 . . 3 (x((x A φ) ψ) ↔ x(x A (φ ψ)))
3 r19.41.1 . . . 4 xψ
4319.41 1554 . . 3 (x((x A φ) ψ) ↔ (x(x A φ) ψ))
52, 4bitr3i 175 . 2 (x(x A (φ ψ)) ↔ (x(x A φ) ψ))
6 df-rex 2286 . 2 (x A (φ ψ) ↔ x(x A (φ ψ)))
7 df-rex 2286 . . 3 (x A φx(x A φ))
87anbi1i 434 . 2 ((x A φ ψ) ↔ (x(x A φ) ψ))
95, 6, 83bitr4i 201 1 (x A (φ ψ) ↔ (x A φ ψ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  Ⅎwnf 1325  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-4 1377  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326  df-rex 2286 This theorem is referenced by:  r19.41v  2440
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