Theorem bnj1340 30148

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

Hypotheses

Ref	Expression
bnj1340.1	⊢ (𝜓 → ∃𝑥𝜃)
bnj1340.2	⊢ (𝜒 ↔ (𝜓 ∧ 𝜃))
bnj1340.3	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
bnj1340	⊢ (𝜓 → ∃𝑥𝜒)

Proof of Theorem bnj1340

Step	Hyp	Ref	Expression
1		bnj1340.3	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2		bnj1340.1	. . 3 ⊢ (𝜓 → ∃𝑥𝜃)
3	1, 2	bnj596 30070	. 2 ⊢ (𝜓 → ∃𝑥(𝜓 ∧ 𝜃))
4		bnj1340.2	. 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃))
5	3, 4	bnj1198 30120	1 ⊢ (𝜓 → ∃𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀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-ex 1696  df-nf 1701

This theorem is referenced by:  bnj1450  30372