Theorem bnj596 31419

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 544	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3		bnj596.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nf5i 2140	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.42 2223	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
6	2, 5	sylibr 226	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828

This theorem is referenced by:  bnj1275  31487  bnj1340  31497  bnj594  31585  bnj1398  31705