Theorem bnj1238 35203

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

Hypothesis

Ref	Expression
bnj1238.1	⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
bnj1238	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Proof of Theorem bnj1238

Step	Hyp	Ref	Expression
1		bnj1238.1	. 2 ⊢ (𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
2		bnj1239 35202	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓)
3	1, 2	sylbi 220	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-rex 3089

This theorem is used by:  bnj1245  35411  bnj1256  35412  bnj1259  35413  bnj1311  35421  bnj1371  35426