Theorem bnj937 31174

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

Hypothesis

Ref	Expression
bnj937.1	⊢ (𝜑 → ∃𝑥𝜓)

Assertion

Ref	Expression
bnj937	⊢ (𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bnj937

Step	Hyp	Ref	Expression
1		bnj937.1	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		19.9v 2064	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	1, 2	sylib 208	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056

This theorem depends on definitions:  df-bi 197  df-ex 1852

This theorem is referenced by:  bnj1265  31215  bnj1379  31233  bnj852  31323  bnj1148  31396  bnj1154  31399  bnj1189  31409  bnj1245  31414  bnj1286  31419  bnj1311  31424  bnj1371  31429  bnj1374  31431  bnj1498  31461  bnj1514  31463