Theorem bnj1345 35454

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

Hypotheses

Ref	Expression
bnj1345.1	⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))
bnj1345.2	⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
bnj1345.3	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
bnj1345	⊢ (𝜑 → ∃𝑥𝜃)

Proof of Theorem bnj1345

Step	Hyp	Ref	Expression
1		bnj1345.1	. . 3 ⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))
2		bnj1345.3	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	bnj1275 35443	. 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
4		bnj1345.2	. 2 ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
5	3, 4	bnj1198 35425	1 ⊢ (𝜑 → ∃𝑥𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1812  df-nf 1816

This theorem is used by:  bnj1379  35460  bnj1521  35481