Theorem r19.41dv 45626

Description: A complex deduction form of r19.41v 3265. (Contributed by Zhi Wang, 6-Sep-2024.)

Hypothesis

Ref	Expression
r19.41dv.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
r19.41dv	⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Distinct variable group:   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.41dv

Step	Hyp	Ref	Expression
1		r19.41dv.1	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2	1	anim1i 617	. 2 ⊢ ((𝜑 ∧ 𝜒) → (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜒))
3		r19.41v 3265	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ 𝜒))
4	2, 3	sylibr 237	1 ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 399  ∃wrex 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-rex 3076

This theorem is referenced by:  opnneilv  45641