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Theorem 19.33b2 1536
Description: The antecedent provides a condition implying the converse of 19.33 1389. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1537 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 657 . . . . 5 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
2 alnex 1404 . . . . . 6 (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
3 alnex 1404 . . . . . 6 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
42, 3orbi12i 691 . . . . 5 ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) ↔ (¬ ∃𝑥𝜓 ∨ ¬ ∃𝑥𝜑))
51, 4bitr4i 180 . . . 4 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) ↔ (∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑))
6 pm2.53 651 . . . . . . 7 ((𝜓𝜑) → (¬ 𝜓𝜑))
76orcoms 659 . . . . . 6 ((𝜑𝜓) → (¬ 𝜓𝜑))
87al2imi 1363 . . . . 5 (∀𝑥(𝜑𝜓) → (∀𝑥 ¬ 𝜓 → ∀𝑥𝜑))
9 pm2.53 651 . . . . . 6 ((𝜑𝜓) → (¬ 𝜑𝜓))
109al2imi 1363 . . . . 5 (∀𝑥(𝜑𝜓) → (∀𝑥 ¬ 𝜑 → ∀𝑥𝜓))
118, 10orim12d 710 . . . 4 (∀𝑥(𝜑𝜓) → ((∀𝑥 ¬ 𝜓 ∨ ∀𝑥 ¬ 𝜑) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
125, 11syl5bi 145 . . 3 (∀𝑥(𝜑𝜓) → ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
1312com12 30 . 2 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
14 19.33 1389 . 2 ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
1513, 14impbid1 134 1 ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie2 1399
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265
This theorem is referenced by:  19.33bdc  1537
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