Theorem bnj1275 35210

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

Hypotheses

Ref	Expression
bnj1275.1	⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))
bnj1275.2	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
bnj1275	⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem bnj1275

Step	Hyp	Ref	Expression
1		bnj1275.2	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bnj1275.1	. . 3 ⊢ (𝜑 → ∃𝑥(𝜓 ∧ 𝜒))
3	1, 2	bnj596 35144	. 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ (𝜓 ∧ 𝜒)))
4		3anass 1110	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bnj1198 35192	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-ex 1809  df-nf 1813

This theorem is used by:  bnj1345  35221  bnj1279  35415