Theorem r19.41 3348

Description: Restricted quantifier version of 19.41 2237. See r19.41v 3347 for a version with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 1-Nov-2010.)

Hypothesis

Ref	Expression
r19.41.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
r19.41	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Proof of Theorem r19.41

Step	Hyp	Ref	Expression
1		anass 471	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
2	1	exbii 1848	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
3		r19.41.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	19.41 2237	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
5	2, 4	bitr3i 279	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
6		df-rex 3144	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
7		df-rex 3144	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8	7	anbi1i 625	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
9	5, 6, 8	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-rex 3144

This theorem is referenced by:  reuxfrdf  30255  iunin1f  30309  2reu8i  43332