Theorem bnj937 32153

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 1988	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	1, 2	sylib 221	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 210  df-ex 1782

This theorem is referenced by:  bnj1265  32194  bnj1379  32212  bnj852  32303  bnj1148  32378  bnj1154  32381  bnj1189  32391  bnj1245  32396  bnj1286  32401  bnj1311  32406  bnj1371  32411  bnj1374  32413  bnj1498  32443  bnj1514  32445