Theorem bj-ceqsalt1 32874

Description: The FOL content of ceqsalt 3228. Lemma for bj-ceqsalt 32875 and bj-ceqsaltv 32876. (TODO: consider removing if it does not add anything to bj-ceqsalt0 32873.) (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 1084	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → ∃𝑥𝜒)
3		biimp 205	. . . . . . 7 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	imim3i 64	. . . . . 6 ⊢ ((𝜒 → (𝜑 ↔ 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
5	4	al2imi 1743	. . . . 5 ⊢ (∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) → (∀𝑥(𝜒 → 𝜑) → ∀𝑥(𝜒 → 𝜓)))
6	5	3ad2ant2 1083	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) → ∀𝑥(𝜒 → 𝜓)))
7		19.23t 2079	. . . . 5 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜒 → 𝜓) ↔ (∃𝑥𝜒 → 𝜓)))
8	7	3ad2ant1 1082	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜓) ↔ (∃𝑥𝜒 → 𝜓)))
9	6, 8	sylibd 229	. . 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 210	. . . . . . 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 1739	. . . 4 ⊢ (∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝜒 → 𝜑)))
15	14	3ad2ant2 1083	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → ∀𝑥(𝜓 → (𝜒 → 𝜑)))
16		19.21t 2073	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝜒 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝜒 → 𝜑))))
17	16	3ad2ant1 1082	. . 3 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜓 → (𝜒 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝜒 → 𝜑))))
18	15, 17	mpbid 222	. 2 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (𝜓 → ∀𝑥(𝜒 → 𝜑)))
19	10, 18	impbid 202	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜒 → (𝜑 ↔ 𝜓)) ∧ 𝜃) → (∀𝑥(𝜒 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)