Theorem bnj1521 32880

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

Hypotheses

Ref	Expression
bnj1521.1	⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)
bnj1521.2	⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))
bnj1521.3	⊢ (𝜒 → ∀𝑥𝜒)

Assertion

Ref	Expression
bnj1521	⊢ (𝜒 → ∃𝑥𝜃)

Proof of Theorem bnj1521

Step	Hyp	Ref	Expression
1		bnj1521.1	. . 3 ⊢ (𝜒 → ∃𝑥 ∈ 𝐵 𝜑)
2	1	bnj1196 32823	. 2 ⊢ (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
3		bnj1521.2	. 2 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑))
4		bnj1521.3	. 2 ⊢ (𝜒 → ∀𝑥𝜒)
5	2, 3, 4	bnj1345 32853	1 ⊢ (𝜒 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087  ∀wal 1537  ∃wex 1779   ∈ wcel 2104  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1089  df-ex 1780  df-nf 1784  df-rex 3072

This theorem is referenced by:  bnj1204  33041  bnj1311  33053  bnj1398  33063  bnj1408  33065  bnj1450  33079  bnj1312  33087  bnj1501  33096  bnj1523  33100