Theorem bnj1521 35548

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 35491	. 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 35521	1 ⊢ (𝜒 → ∃𝑥𝜃)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103  ∀wal 1568  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-nf 1817  df-rex 3091

This theorem is used by:  bnj1204  35709  bnj1311  35721  bnj1398  35731  bnj1408  35733  bnj1450  35747  bnj1312  35755  bnj1501  35764  bnj1523  35768