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Theorem bj-ceqsalt1 32521
Description: The FOL content of ceqsalt 3214. Lemma for bj-ceqsalt 32522 and bj-ceqsaltv 32523. (TODO: consider removing if it does not add anything to bj-ceqsalt0 32520.) (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 1082 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → ∃𝑥𝜒)
3 biimp 205 . . . . . . 7 ((𝜑𝜓) → (𝜑𝜓))
43imim3i 64 . . . . . 6 ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓)))
54al2imi 1740 . . . . 5 (∀𝑥(𝜒 → (𝜑𝜓)) → (∀𝑥(𝜒𝜑) → ∀𝑥(𝜒𝜓)))
653ad2ant2 1081 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → ∀𝑥(𝜒𝜓)))
7 19.23t 2077 . . . . 5 (Ⅎ𝑥𝜓 → (∀𝑥(𝜒𝜓) ↔ (∃𝑥𝜒𝜓)))
873ad2ant1 1080 . . . 4 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜓) ↔ (∃𝑥𝜒𝜓)))
96, 8sylibd 229 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))
102, 9mpid 44 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) → 𝜓))
11 biimpr 210 . . . . . . 7 ((𝜑𝜓) → (𝜓𝜑))
1211imim2i 16 . . . . . 6 ((𝜒 → (𝜑𝜓)) → (𝜒 → (𝜓𝜑)))
1312com23 86 . . . . 5 ((𝜒 → (𝜑𝜓)) → (𝜓 → (𝜒𝜑)))
1413alimi 1736 . . . 4 (∀𝑥(𝜒 → (𝜑𝜓)) → ∀𝑥(𝜓 → (𝜒𝜑)))
15143ad2ant2 1081 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → ∀𝑥(𝜓 → (𝜒𝜑)))
16 19.21t 2071 . . . 4 (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝜒𝜑)) ↔ (𝜓 → ∀𝑥(𝜒𝜑))))
17163ad2ant1 1080 . . 3 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜓 → (𝜒𝜑)) ↔ (𝜓 → ∀𝑥(𝜒𝜑))))
1815, 17mpbid 222 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (𝜓 → ∀𝑥(𝜒𝜑)))
1910, 18impbid 202 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑𝜓)) ∧ 𝜃) → (∀𝑥(𝜒𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wal 1478  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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