Theorem 19.34 1614

Description: Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.34	⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))

Proof of Theorem 19.34

Step	Hyp	Ref	Expression
1		19.2 1569	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2	1	orim1i 709	. 2 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
3		19.43 1559	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
4	2, 3	sylibr 132	1 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 661  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)