Theorem r19.43 2623

Description: Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)

Assertion

Ref	Expression
r19.43	⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.43

Step	Hyp	Ref	Expression
1		df-rex 2449	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓)))
2		andi 808	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓)))
3	2	exbii 1593	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝜑 ∨ 𝜓)) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓)))
4	1, 3	bitri 183	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓)))
5		19.43 1616	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∨ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
6	4, 5	bitri 183	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∨ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
7		df-rex 2449	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 2449	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
9	7, 8	orbi12i 754	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∨ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
10	6, 9	bitr4i 186	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 698  ∃wex 1480   ∈ wcel 2136  ∃wrex 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-rex 2449

This theorem is referenced by:  r19.44av  2624  r19.45av  2625  r19.45mv  3501  r19.44mv  3502  iunun  3943  ltexprlemloc  7544  pythagtriplem2  12194  pythagtrip  12211