Theorem bnj1345 30149

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 30138	. 2 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
4		bnj1345.2	. 2 ⊢ (𝜃 ↔ (𝜑 ∧ 𝜓 ∧ 𝜒))
5	3, 4	bnj1198 30120	1 ⊢ (𝜑 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-nf 1701

This theorem is referenced by:  bnj1379  30155  bnj1521  30175