Theorem bnj115 32604

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj115.1	⊢ (𝜂 ↔ ∀𝑛 ∈ 𝐷 (𝜏 → 𝜃))

Assertion

Ref	Expression
bnj115	⊢ (𝜂 ↔ ∀𝑛((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Proof of Theorem bnj115

Step	Hyp	Ref	Expression
1		bnj115.1	. 2 ⊢ (𝜂 ↔ ∀𝑛 ∈ 𝐷 (𝜏 → 𝜃))
2		df-ral 3068	. 2 ⊢ (∀𝑛 ∈ 𝐷 (𝜏 → 𝜃) ↔ ∀𝑛(𝑛 ∈ 𝐷 → (𝜏 → 𝜃)))
3		impexp 450	. . . 4 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃) ↔ (𝑛 ∈ 𝐷 → (𝜏 → 𝜃)))
4	3	bicomi 223	. . 3 ⊢ ((𝑛 ∈ 𝐷 → (𝜏 → 𝜃)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))
5	4	albii 1823	. 2 ⊢ (∀𝑛(𝑛 ∈ 𝐷 → (𝜏 → 𝜃)) ↔ ∀𝑛((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))
6	1, 2, 5	3bitri 296	1 ⊢ (𝜂 ↔ ∀𝑛((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   ∈ wcel 2108  ∀wral 3063

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-ral 3068

This theorem is referenced by:  bnj953  32819  bnj964  32823  bnj1090  32859  bnj1112  32863