Theorem 19.34 1574

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 1529	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2	1	orim1i 677	. 2 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → (∃𝑥𝜑 ∨ ∃𝑥𝜓))
3		19.43 1519	. 2 ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
4	2, 3	sylibr 137	1 ⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 629  ∀wal 1241  ∃wex 1381

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-io 630  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)