Theorem bnj1521 35209

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 35152	. 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 35182	1 ⊢ (𝜒 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1101  ∀wal 1566  ∃wex 1807   ∈ wcel 2150  ∃wrex 3096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2183  ax-12 2220

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-ex 1808  df-nf 1812  df-rex 3097

This theorem is referenced by:  bnj1204  35370  bnj1311  35382  bnj1398  35392  bnj1408  35394  bnj1450  35408  bnj1312  35416  bnj1501  35425  bnj1523  35429