Theorem bnj1340 35453

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 35377	. 2 ⊢ (𝜓 → ∃𝑥(𝜓 ∧ 𝜃))
4		bnj1340.2	. 2 ⊢ (𝜒 ↔ (𝜓 ∧ 𝜃))
5	3, 4	bnj1198 35425	1 ⊢ (𝜓 → ∃𝑥𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816

This theorem is used by:  bnj1450  35680