Theorem bnj1238 30131

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 30130	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓)
3	1, 2	sylbi 206	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902

This theorem is referenced by:  bnj1245  30336  bnj1256  30337  bnj1259  30338  bnj1311  30346  bnj1371  30351