Theorem bnj596 30816

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 574	. 2 ⊢ (𝜑 → (𝜑 ∧ ∃𝑥𝜓))
3		bnj596.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	nf5i 2024	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.42-1 2104	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
6	2, 5	syl 17	1 ⊢ (𝜑 → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj1275  30884  bnj1340  30894  bnj594  30982  bnj1398  31102