Theorem 19.34 2025

Description: Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Assertion

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

Proof of Theorem 19.34

Step	Hyp	Ref	Expression
1		19.2 2009	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2	1	orim1i 923	. 2 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
3		19.43 1915	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
4	2, 3	sylibr 237	1 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-6 2000

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813

This theorem is used by: (None)