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Theorem 19.33b 1608
 Description: The antecedent provides a condition implying the converse of 19.33 1607. Compare Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
Assertion
Ref Expression
19.33b (¬ (xφ xψ) → (x(φ ψ) ↔ (xφ xψ)))

Proof of Theorem 19.33b
StepHypRef Expression
1 ianor 474 . . 3 (¬ (xφ xψ) ↔ (¬ xφ ¬ xψ))
2 alnex 1543 . . . . . 6 (x ¬ φ ↔ ¬ xφ)
3 pm2.53 362 . . . . . . 7 ((φ ψ) → (¬ φψ))
43al2imi 1561 . . . . . 6 (x(φ ψ) → (x ¬ φxψ))
52, 4syl5bir 209 . . . . 5 (x(φ ψ) → (¬ xφxψ))
6 olc 373 . . . . 5 (xψ → (xφ xψ))
75, 6syl6com 31 . . . 4 xφ → (x(φ ψ) → (xφ xψ)))
8 19.30 1604 . . . . . . 7 (x(φ ψ) → (xφ xψ))
98orcomd 377 . . . . . 6 (x(φ ψ) → (xψ xφ))
109ord 366 . . . . 5 (x(φ ψ) → (¬ xψxφ))
11 orc 374 . . . . 5 (xφ → (xφ xψ))
1210, 11syl6com 31 . . . 4 xψ → (x(φ ψ) → (xφ xψ)))
137, 12jaoi 368 . . 3 ((¬ xφ ¬ xψ) → (x(φ ψ) → (xφ xψ)))
141, 13sylbi 187 . 2 (¬ (xφ xψ) → (x(φ ψ) → (xφ xψ)))
15 19.33 1607 . 2 ((xφ xψ) → x(φ ψ))
1614, 15impbid1 194 1 (¬ (xφ xψ) → (x(φ ψ) ↔ (xφ xψ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1542 This theorem is referenced by: (None)
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