Theorem rexiunxp

Description: Write a double restricted quantification as one universal quantifier. In this version of rexxp , B ( y ) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015)

	Ref	Expression
Hypothesis	ralxp.1	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexiunxp	⊢ ( ∃ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralxp.1	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( 𝑥 = ⟨ 𝑦 , 𝑧 ⟩ → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	raliunxp	⊢ ( ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) ¬ 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 ¬ 𝜓 )
4		ralnex	⊢ ( ∀ 𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃ 𝑧 ∈ 𝐵 𝜓 )
5	4	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀ 𝑦 ∈ 𝐴 ¬ ∃ 𝑧 ∈ 𝐵 𝜓 )
6	3 5	bitri	⊢ ( ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) ¬ 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 ¬ ∃ 𝑧 ∈ 𝐵 𝜓 )
7	6	notbii	⊢ ( ¬ ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) ¬ 𝜑 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ ∃ 𝑧 ∈ 𝐵 𝜓 )
8		dfrex2	⊢ ( ∃ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) 𝜑 ↔ ¬ ∀ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) ¬ 𝜑 )
9		dfrex2	⊢ ( ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀ 𝑦 ∈ 𝐴 ¬ ∃ 𝑧 ∈ 𝐵 𝜓 )
10	7 8 9	3bitr4i	⊢ ( ∃ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × 𝐵 ) 𝜑 ↔ ∃ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ 𝐵 𝜓 )