Theorem bnj534 35137

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 1978	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
3	1, 2	sylibr 237	1 ⊢ (𝜒 → ∃𝑥(𝜑 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809

This theorem is used by:  bnj600  35316  bnj852  35318