Theorem bnj937 35268

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 2017	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	1, 2	sylib 221	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  bnj1265  35308  bnj1379  35326  bnj852  35417  bnj1148  35492  bnj1154  35495  bnj1189  35505  bnj1245  35510  bnj1286  35515  bnj1311  35520  bnj1371  35525  bnj1374  35527  bnj1498  35557  bnj1514  35559