Theorem bnj596 32705

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 548	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3		bnj596.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nf5i 2145	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.42 2232	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
6	2, 5	sylibr 233	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-nf 1790

This theorem is referenced by:  bnj1275  32772  bnj1340  32782  bnj594  32871  bnj1398  32993