Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2r19.29 Structured version   Visualization version   GIF version

Theorem 2r19.29 3217
 Description: Theorem r19.29 3210 with two quantifiers. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
2r19.29 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))

Proof of Theorem 2r19.29
StepHypRef Expression
1 r19.29 3210 . 2 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓))
2 r19.29 3210 . . 3 ((∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓) → ∃𝑦𝐵 (𝜑𝜓))
32reximi 3149 . 2 (∃𝑥𝐴 (∀𝑦𝐵 𝜑 ∧ ∃𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
41, 3syl 17 1 ((∀𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑥𝐴𝑦𝐵 𝜓) → ∃𝑥𝐴𝑦𝐵 (𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wral 3050  ∃wrex 3051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-ral 3055  df-rex 3056 This theorem is referenced by:  prter2  34688
 Copyright terms: Public domain W3C validator