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Theorem 19.41h 1572
 Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1573 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.41h.1 (ψxψ)
Assertion
Ref Expression
19.41h (x(φ ψ) ↔ (xφ ψ))

Proof of Theorem 19.41h
StepHypRef Expression
1 19.40 1519 . . 3 (x(φ ψ) → (xφ xψ))
2 19.41h.1 . . . . 5 (ψxψ)
3 id 19 . . . . 5 (ψψ)
42, 3exlimih 1481 . . . 4 (xψψ)
54anim2i 324 . . 3 ((xφ xψ) → (xφ ψ))
61, 5syl 14 . 2 (x(φ ψ) → (xφ ψ))
7 pm3.21 251 . . . 4 (ψ → (φ → (φ ψ)))
82, 7eximdh 1499 . . 3 (ψ → (xφx(φ ψ)))
98impcom 116 . 2 ((xφ ψ) → x(φ ψ))
106, 9impbii 117 1 (x(φ ψ) ↔ (xφ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.42h  1574  sbh  1656  sbidm  1728  19.41v  1779  2exeu  1989
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