Theorem dftr2c

Description: Variant of dftr2 with commuted quantifiers, useful for shortening proofs and avoiding ax-11 . (Contributed by BTernaryTau, 28-Dec-2024)

	Ref	Expression
Assertion	dftr2c	⊢ ( Tr 𝐴 ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		dftr2	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
2		elequ1	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦 ) )
3	2	anbi1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ) )
4		eleq1w	⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) )
5	3 4	imbi12d	⊢ ( 𝑥 = 𝑧 → ( ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑧 ∈ 𝐴 ) ) )
6		elequ2	⊢ ( 𝑦 = 𝑧 → ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧 ) )
7		eleq1w	⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) )
8	6 7	anbi12d	⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴 ) ) )
9	8	imbi1d	⊢ ( 𝑦 = 𝑧 → ( ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ↔ ( ( 𝑥 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ) )
10	5 9	alcomw	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )
11	1 10	bitri	⊢ ( Tr 𝐴 ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) )