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Theorem 19.33b2 1591
Description: The antecedent provides a condition implying the converse of 19.33 1443. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1592 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.33b2 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))

Proof of Theorem 19.33b2
StepHypRef Expression
1 orcom 700 . . . . 5 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
2 alnex 1458 . . . . . 6 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
3 alnex 1458 . . . . . 6 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
42, 3orbi12i 736 . . . . 5 ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
51, 4bitr4i 186 . . . 4 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑))
6 pm2.53 694 . . . . . . 7 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 702 . . . . . 6 ((𝜑𝜓) → (¬ 𝜓𝜑))
87al2imi 1417 . . . . 5 (∀𝑥(𝜑𝜓) → (∀𝑥 ¬ 𝜓 → ∀𝑥𝜑))
9 pm2.53 694 . . . . . 6 ((𝜑𝜓) → (¬ 𝜑𝜓))
109al2imi 1417 . . . . 5 (∀𝑥(𝜑𝜓) → (∀𝑥 ¬ 𝜑 → ∀𝑥𝜓))
118, 10orim12d 758 . . . 4 (∀𝑥(𝜑𝜓) → ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
125, 11syl5bi 151 . . 3 (∀𝑥(𝜑𝜓) → ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
1312com12 30 . 2 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
14 19.33 1443 . 2 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
1513, 14impbid1 141 1 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680  wal 1312  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by:  19.33bdc  1592
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