Theorem bnj1345 32804

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 32793	. 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
4		bnj1345.2	. 2 ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
5	3, 4	bnj1198 32775	1 ⊢ (𝜑 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-ex 1783  df-nf 1787

This theorem is referenced by:  bnj1379  32810  bnj1521  32831