Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ceqsalt1 Structured version   Visualization version   GIF version

Theorem bj-ceqsalt1 33231
Description: The FOL content of ceqsalt 3381. Lemma for bj-ceqsalt 33232 and bj-ceqsaltv 33233. (TODO: consider removing if it does not add anything to bj-ceqsalt0 33230.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-ceqsalt1.1 (𝜃 → ∃𝑥𝜒)
Assertion
Ref Expression
bj-ceqsalt1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))

Proof of Theorem bj-ceqsalt1
StepHypRef Expression
1 bj-ceqsalt1.1 . . . 4 (𝜃 → ∃𝑥𝜒)
213ad2ant3 1165 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → ∃𝑥𝜒)
3 biimp 206 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
43imim3i 64 . . . . . 6 ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓)))
54al2imi 1910 . . . . 5 (∀𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥(𝜒𝜑) → ∀𝑥(𝜒𝜓)))
653ad2ant2 1164 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → ∀𝑥(𝜒𝜓)))
7 19.23t 2242 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝜒𝜓) ↔ (∃𝑥𝜒𝜓)))
873ad2ant1 1163 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜓) ↔ (∃𝑥𝜒𝜓)))
96, 8sylibd 230 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))
102, 9mpid 44 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → 𝜓))
11 biimpr 211 . . . . . . 7 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 16 . . . . . 6 ((𝜒 → (𝜑𝜓)) → (𝜒 → (𝜓𝜑)))
1312com23 86 . . . . 5 ((𝜒 → (𝜑𝜓)) → (𝜓 → (𝜒𝜑)))
1413alimi 1906 . . . 4 (∀𝑥(𝜒 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝜒𝜑)))
15143ad2ant2 1164 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → ∀𝑥(𝜓 → (𝜒𝜑)))
16 19.21t 2238 . . . 4 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝜒𝜑)) ↔ (𝜓 → ∀𝑥(𝜒𝜑))))
17163ad2ant1 1163 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜓 → (𝜒𝜑)) ↔ (𝜓 → ∀𝑥(𝜒𝜑))))
1815, 17mpbid 223 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (𝜓 → ∀𝑥(𝜒𝜑)))
1910, 18impbid 203 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107  wal 1650  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-3an 1109  df-ex 1875  df-nf 1879
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator