Theorem 19.40-2 1620

Description: Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
19.40-2	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))

Proof of Theorem 19.40-2

Step	Hyp	Ref	Expression
1		19.40 1619	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) → (∃𝑦𝜑 ∧ ∃𝑦𝜓))
2	1	eximi 1588	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓))
3		19.40 1619	. 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ ∃𝑦𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))
4	2, 3	syl 14	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1480

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)