Theorem bnj534 32003

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

Hypothesis

Ref	Expression
bnj534.1	⊢ (𝜒 → (∃𝑥𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj534	⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem bnj534

Step	Hyp	Ref	Expression
1		bnj534.1	. 2 ⊢ (𝜒 → (∃𝑥𝜑 ∧ 𝜓))
2		19.41v 1943	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
3	1, 2	sylibr 236	1 ⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774

This theorem is referenced by:  bnj600  32184  bnj852  32186