Theorem bnj1275 34789

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 34722	. 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ (𝜓 ∧ 𝜒)))
4		3anass 1095	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
5	3, 4	bnj1198 34771	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1535  ∃wex 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1778  df-nf 1782

This theorem is referenced by:  bnj1345  34800  bnj1279  34994