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Metamath Proof Explorer

Theorem 3reeanv

Description: Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010)

		Ref	Expression
	Assertion	3reeanv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.41v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) ↔ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
2		reeanv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )
3	1 2	bianbi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) ↔ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
4		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
5	4	2rexbii	⊢ ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
6		reeanv	⊢ ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
7	5 6	bitri	⊢ ( ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
8	7	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ∈ 𝐴 ( ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
9		df-3an	⊢ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) ↔ ( ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )
10	3 8 9	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐶 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑧 ∈ 𝐶 𝜒 ) )