Theorem bj-ceqsalt1

Description: The FOL content of ceqsalt . Lemma for bj-ceqsalt and bj-ceqsaltv . TODO: consider removing if it does not add anything to bj-ceqsalt0 . (Contributed by BJ, 26-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-ceqsalt1.1	⊢ ( 𝜃 → ∃ 𝑥 𝜒 )
Assertion	bj-ceqsalt1	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalt1.1	⊢ ( 𝜃 → ∃ 𝑥 𝜒 )
2	1	3ad2ant3	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ∃ 𝑥 𝜒 )
3		biimp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
4	3	imim3i	⊢ ( ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝜒 → 𝜑 ) → ( 𝜒 → 𝜓 ) ) )
5	4	al2imi	⊢ ( ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ∀ 𝑥 ( 𝜒 → 𝜓 ) ) )
6	5	3ad2ant2	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ∀ 𝑥 ( 𝜒 → 𝜓 ) ) )
7		19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜒 → 𝜓 ) ↔ ( ∃ 𝑥 𝜒 → 𝜓 ) ) )
8	7	3ad2ant1	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜓 ) ↔ ( ∃ 𝑥 𝜒 → 𝜓 ) ) )
9	6 8	sylibd	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → ( ∃ 𝑥 𝜒 → 𝜓 ) ) )
10	2 9	mpid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) → 𝜓 ) )
11		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
12	11	imim2i	⊢ ( ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜒 → ( 𝜓 → 𝜑 ) ) )
13	12	com23	⊢ ( ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ( 𝜒 → 𝜑 ) ) )
14	13	alimi	⊢ ( ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) → ∀ 𝑥 ( 𝜓 → ( 𝜒 → 𝜑 ) ) )
15	14	3ad2ant2	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ∀ 𝑥 ( 𝜓 → ( 𝜒 → 𝜑 ) ) )
16		19.21t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜓 → ( 𝜒 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝜒 → 𝜑 ) ) ) )
17	16	3ad2ant1	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜓 → ( 𝜒 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝜒 → 𝜑 ) ) ) )
18	15 17	mpbid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( 𝜓 → ∀ 𝑥 ( 𝜒 → 𝜑 ) ) )
19	10 18	impbid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜒 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜃 ) → ( ∀ 𝑥 ( 𝜒 → 𝜑 ) ↔ 𝜓 ) )