Theorem bnj596 30070

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

Hypotheses

Ref	Expression
bnj596.1	⊢ (𝜑 → ∀𝑥𝜑)
bnj596.2	⊢ (𝜑 → ∃𝑥𝜓)

Assertion

Ref	Expression
bnj596	⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem bnj596

Step	Hyp	Ref	Expression
1		bnj596.2	. . 3 ⊢ (𝜑 → ∃𝑥𝜓)
2	1	ancli 572	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3		bnj596.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nf5i 2011	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.42-1 2091	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
6	2, 5	syl 17	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  bnj1275  30138  bnj1340  30148  bnj594  30236  bnj1398  30356