Theorem bnj1019

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

	Ref	Expression
Assertion	bnj1019	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		19.42v	⊢ ( ∃ 𝑝 ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ) ∧ 𝜏 ) ↔ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ) ∧ ∃ 𝑝 𝜏 ) )
2		bnj258	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ) ∧ 𝜏 ) )
3	2	exbii	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ∃ 𝑝 ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ) ∧ 𝜏 ) )
4		df-bnj17	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ↔ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ) ∧ ∃ 𝑝 𝜏 ) )
5	1 3 4	3bitr4i	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )