Theorem bnj1265 32077

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

Hypothesis

Ref	Expression
bnj1265.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj1265	⊢ (𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem bnj1265

Step	Hyp	Ref	Expression
1		bnj1265.1	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2	1	bnj1196 32059	. . 3 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	bnj1266 32076	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4	3	bnj937 32036	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃wrex 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-rex 3142

This theorem is referenced by:  bnj1253  32282  bnj1280  32285  bnj1296  32286  bnj1371  32294  bnj1497  32325