Theorem bj-ceqsalt0

Description: The FOL content of ceqsalt . Lemma for bj-ceqsalt and bj-ceqsaltv . (Contributed by BJ, 26-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ceqsalt0	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		simp3	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ∃ 𝑥 𝜃 )
2		biimp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
3	2	imim3i	⊢ ( ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜃 → 𝜓 ) ) )
4	3	al2imi	⊢ ( ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) → ∀ 𝑥 ( 𝜃 → 𝜓 ) ) )
5	4	3ad2ant2	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) → ∀ 𝑥 ( 𝜃 → 𝜓 ) ) )
6		19.23t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜃 → 𝜓 ) ↔ ( ∃ 𝑥 𝜃 → 𝜓 ) ) )
7	6	3ad2ant1	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜓 ) ↔ ( ∃ 𝑥 𝜃 → 𝜓 ) ) )
8	5 7	sylibd	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) → ( ∃ 𝑥 𝜃 → 𝜓 ) ) )
9	1 8	mpid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) → 𝜓 ) )
10		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
11	10	imim2i	⊢ ( ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜃 → ( 𝜓 → 𝜑 ) ) )
12	11	com23	⊢ ( ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ( 𝜃 → 𝜑 ) ) )
13	12	alimi	⊢ ( ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) → ∀ 𝑥 ( 𝜓 → ( 𝜃 → 𝜑 ) ) )
14	13	3ad2ant2	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ∀ 𝑥 ( 𝜓 → ( 𝜃 → 𝜑 ) ) )
15		19.21t	⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝜓 → ( 𝜃 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝜃 → 𝜑 ) ) ) )
16	15	3ad2ant1	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜓 → ( 𝜃 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝜃 → 𝜑 ) ) ) )
17	14 16	mpbid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( 𝜓 → ∀ 𝑥 ( 𝜃 → 𝜑 ) ) )
18	9 17	impbid	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝜃 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∃ 𝑥 𝜃 ) → ( ∀ 𝑥 ( 𝜃 → 𝜑 ) ↔ 𝜓 ) )