Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  reximd2a Structured version   Visualization version   GIF version

Theorem reximd2a 3151
 Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)
Hypotheses
Ref Expression
reximd2a.1 𝑥𝜑
reximd2a.2 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)
reximd2a.3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
reximd2a.4 (𝜑 → ∃𝑥𝐴 𝜓)
Assertion
Ref Expression
reximd2a (𝜑 → ∃𝑥𝐵 𝜒)

Proof of Theorem reximd2a
StepHypRef Expression
1 reximd2a.4 . 2 (𝜑 → ∃𝑥𝐴 𝜓)
2 reximd2a.1 . . . 4 𝑥𝜑
3 reximd2a.2 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥𝐵)
4 reximd2a.3 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
53, 4jca 555 . . . . 5 (((𝜑𝑥𝐴) ∧ 𝜓) → (𝑥𝐵𝜒))
65expl 649 . . . 4 (𝜑 → ((𝑥𝐴𝜓) → (𝑥𝐵𝜒)))
72, 6eximd 2232 . . 3 (𝜑 → (∃𝑥(𝑥𝐴𝜓) → ∃𝑥(𝑥𝐵𝜒)))
8 df-rex 3056 . . 3 (∃𝑥𝐴 𝜓 ↔ ∃𝑥(𝑥𝐴𝜓))
9 df-rex 3056 . . 3 (∃𝑥𝐵 𝜒 ↔ ∃𝑥(𝑥𝐵𝜒))
107, 8, 93imtr4g 285 . 2 (𝜑 → (∃𝑥𝐴 𝜓 → ∃𝑥𝐵 𝜒))
111, 10mpd 15 1 (𝜑 → ∃𝑥𝐵 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1853  Ⅎwnf 1857   ∈ wcel 2139  ∃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  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-nf 1859  df-rex 3056 This theorem is referenced by:  locfinreflem  30237
 Copyright terms: Public domain W3C validator