Theorem bnj1171

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1171.13	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐵 ⊆ 𝐴 )
	bnj1171.129	⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
Assertion	bnj1171	⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1171.13	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐵 ⊆ 𝐴 )
2		bnj1171.129	⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
3	1	sseld	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑤 ∈ 𝐵 → 𝑤 ∈ 𝐴 ) )
4	3	pm4.71rd	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑤 ∈ 𝐵 ↔ ( 𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) ) )
5	4	imbi1d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ↔ ( ( 𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) → ¬ 𝑤 𝑅 𝑧 ) ) )
6		impexp	⊢ ( ( ( 𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵 ) → ¬ 𝑤 𝑅 𝑧 ) ↔ ( 𝑤 ∈ 𝐴 → ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
7	5 6	bitrdi	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ↔ ( 𝑤 ∈ 𝐴 → ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) )
8		con2b	⊢ ( ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ↔ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) )
9	8	imbi2i	⊢ ( ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ↔ ( 𝑤 ∈ 𝐴 → ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )
10	7 9	bitr4di	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ↔ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) )
11	10	anbi2d	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
12	11	pm5.74i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
13	12	albii	⊢ ( ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) ↔ ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
14	13	exbii	⊢ ( ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) ) ↔ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐴 → ( 𝑤 𝑅 𝑧 → ¬ 𝑤 ∈ 𝐵 ) ) ) ) )
15	2 14	mpbir	⊢ ∃ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑧 ∈ 𝐵 ∧ ( 𝑤 ∈ 𝐵 → ¬ 𝑤 𝑅 𝑧 ) ) )