Theorem r19.41 2523

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (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 394	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
2	1	exbii 1542	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
3		r19.41.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	19.41 1622	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
5	2, 4	bitr3i 185	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
6		df-rex 2366	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∧ 𝜓)))
7		df-rex 2366	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8	7	anbi1i 447	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ 𝜓))
9	5, 6, 8	3bitr4i 211	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104  Ⅎwnf 1395  ∃wex 1427   ∈ wcel 1439  ∃wrex 2361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-rex 2366

This theorem is referenced by:  r19.41v  2524