Theorem bj-ceqsalt1 34997

Description: The FOL content of ceqsalt 3452. Lemma for bj-ceqsalt 34998 and bj-ceqsaltv 34999. TODO: consider removing if it does not add anything to bj-ceqsalt0 34996. (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 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Proof of Theorem bj-ceqsalt1

Step	Hyp	Ref	Expression
1		bj-ceqsalt1.1	. . . 4 ⊢ (𝜃 → ∃𝑥𝜒)
2	1	3ad2ant3 1133	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → ∃𝑥𝜒)
3		biimp 214	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imim3i 64	. . . . . 6 ⊢ ((𝜒 → (𝜑 ↔ 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
5	4	al2imi 1819	. . . . 5 ⊢ (∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝜒 → 𝜑) → ∀𝑥(𝜒 → 𝜓)))
6	5	3ad2ant2 1132	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) → ∀𝑥(𝜒 → 𝜓)))
7		19.23t 2206	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜒 → 𝜓) ↔ (∃𝑥𝜒 → 𝜓)))
8	7	3ad2ant1 1131	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜓) ↔ (∃𝑥𝜒 → 𝜓)))
9	6, 8	sylibd 238	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) → (∃𝑥𝜒 → 𝜓)))
10	2, 9	mpid 44	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) → 𝜓))
11		biimpr 219	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
12	11	imim2i 16	. . . . . 6 ⊢ ((𝜒 → (𝜑 ↔ 𝜓)) → (𝜒 → (𝜓 → 𝜑)))
13	12	com23 86	. . . . 5 ⊢ ((𝜒 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝜒 → 𝜑)))
14	13	alimi 1815	. . . 4 ⊢ (∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝜒 → 𝜑)))
15	14	3ad2ant2 1132	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → ∀𝑥(𝜓 → (𝜒 → 𝜑)))
16		19.21t 2202	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝜒 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝜒 → 𝜑))))
17	16	3ad2ant1 1131	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜓 → (𝜒 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝜒 → 𝜑))))
18	15, 17	mpbid 231	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (𝜓 → ∀𝑥(𝜒 → 𝜑)))
19	10, 18	impbid 211	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)