Theorem bnj1019 32738

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

Assertion

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

Distinct variable groups:   𝜒,𝑝   𝜂,𝑝   𝜃,𝑝

Allowed substitution hint:   𝜏(𝑝)

Proof of Theorem bnj1019

Step	Hyp	Ref	Expression
1		19.42v 1960	. 2 ⊢ (∃𝑝((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏))
2		bnj258 32666	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏))
3	2	exbii 1853	. 2 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ∃𝑝((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏))
4		df-bnj17 32645	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏))
5	1, 3, 4	3bitr4i 302	1 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∃wex 1785   ∧ w-bnj17 32644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ex 1786  df-bnj17 32645

This theorem is referenced by:  bnj1018g  32922  bnj1018  32923  bnj1020  32924  bnj1021  32925