Theorem bnj1521 30921

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 30865	. 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 30895	1 ⊢ (𝜒 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037  ∀wal 1481  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  bnj1204  31080  bnj1311  31092  bnj1398  31102  bnj1408  31104  bnj1450  31118  bnj1312  31126  bnj1501  31135  bnj1523  31139