Theorem bnj1521 32810

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 32753	. 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 32783	1 ⊢ (𝜒 → ∃𝑥𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085  ∀wal 1539  ∃wex 1785   ∈ wcel 2109  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ex 1786  df-nf 1790  df-rex 3071

This theorem is referenced by:  bnj1204  32971  bnj1311  32983  bnj1398  32993  bnj1408  32995  bnj1450  33009  bnj1312  33017  bnj1501  33026  bnj1523  33030