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Theorem 19.33b2 1517
Description: The antecedent provides a condition implying the converse of 19.33 1370. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1518 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.33b2 ((¬ xφ ¬ xψ) → (x(φ ψ) ↔ (xφ xψ)))

Proof of Theorem 19.33b2
StepHypRef Expression
1 orcom 646 . . . . 5 ((¬ xφ ¬ xψ) ↔ (¬ xψ ¬ xφ))
2 alnex 1385 . . . . . 6 (x ¬ ψ ↔ ¬ xψ)
3 alnex 1385 . . . . . 6 (x ¬ φ ↔ ¬ xφ)
42, 3orbi12i 680 . . . . 5 ((x ¬ ψ x ¬ φ) ↔ (¬ xψ ¬ xφ))
51, 4bitr4i 176 . . . 4 ((¬ xφ ¬ xψ) ↔ (x ¬ ψ x ¬ φ))
6 pm2.53 640 . . . . . . 7 ((ψ φ) → (¬ ψφ))
76orcoms 648 . . . . . 6 ((φ ψ) → (¬ ψφ))
87al2imi 1344 . . . . 5 (x(φ ψ) → (x ¬ ψxφ))
9 pm2.53 640 . . . . . 6 ((φ ψ) → (¬ φψ))
109al2imi 1344 . . . . 5 (x(φ ψ) → (x ¬ φxψ))
118, 10orim12d 699 . . . 4 (x(φ ψ) → ((x ¬ ψ x ¬ φ) → (xφ xψ)))
125, 11syl5bi 141 . . 3 (x(φ ψ) → ((¬ xφ ¬ xψ) → (xφ xψ)))
1312com12 27 . 2 ((¬ xφ ¬ xψ) → (x(φ ψ) → (xφ xψ)))
14 19.33 1370 . 2 ((xφ xψ) → x(φ ψ))
1513, 14impbid1 130 1 ((¬ xφ ¬ xψ) → (x(φ ψ) ↔ (xφ xψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie2 1380
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248
This theorem is referenced by:  19.33bdc  1518
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