Theorem bnj1019 35477

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 1988	. 2 ⊢ (∃𝑝((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏))
2		bnj258 35406	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏))
3	2	exbii 1881	. 2 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ∃𝑝((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏))
4		df-bnj17 35385	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏))
5	1, 3, 4	3bitr4i 306	1 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∃wex 1812   ∧ w-bnj17 35384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-bnj17 35385

This theorem is used by:  bnj1018g  35660  bnj1018  35661  bnj1020  35662  bnj1021  35663