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Theorem r19.43 2462
 Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
Assertion
Ref Expression
r19.43 (x A (φ ψ) ↔ (x A φ x A ψ))

Proof of Theorem r19.43
StepHypRef Expression
1 df-rex 2306 . . . 4 (x A (φ ψ) ↔ x(x A (φ ψ)))
2 andi 730 . . . . 5 ((x A (φ ψ)) ↔ ((x A φ) (x A ψ)))
32exbii 1493 . . . 4 (x(x A (φ ψ)) ↔ x((x A φ) (x A ψ)))
41, 3bitri 173 . . 3 (x A (φ ψ) ↔ x((x A φ) (x A ψ)))
5 19.43 1516 . . 3 (x((x A φ) (x A ψ)) ↔ (x(x A φ) x(x A ψ)))
64, 5bitri 173 . 2 (x A (φ ψ) ↔ (x(x A φ) x(x A ψ)))
7 df-rex 2306 . . 3 (x A φx(x A φ))
8 df-rex 2306 . . 3 (x A ψx(x A ψ))
97, 8orbi12i 680 . 2 ((x A φ x A ψ) ↔ (x(x A φ) x(x A ψ)))
106, 9bitr4i 176 1 (x A (φ ψ) ↔ (x A φ x A ψ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∨ wo 628  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301 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-io 629  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110  df-rex 2306 This theorem is referenced by:  r19.44av  2463  r19.45av  2464  r19.45mv  3309  iunun  3725  ltexprlemloc  6580
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