Theorem bj-rexcom4 32869

Description: Remove from rexcom4 3225 dependency on ax-ext 2602 and ax-13 2246 (and on df-or 385, df-tru 1486, df-sb 1881, df-clab 2609, df-cleq 2615, df-clel 2618, df-nfc 2753, df-v 3202). This proof uses only df-rex 2918 on top of first-order logic. (Contributed by BJ, 13-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-rexcom4	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem bj-rexcom4

Step	Hyp	Ref	Expression
1		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
2		19.42v 1918	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
3	2	bicomi 214	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
4	3	exbii 1774	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
5		excom 2042	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rex 2918	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	bicomi 214	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)
8	7	exbii 1774	. . . 4 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
9	5, 8	bitri 264	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
10	4, 9	bitri 264	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
11	1, 10	bitri 264	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913

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-11 2034

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-rex 2918

This theorem is referenced by:  bj-rexcom4a  32870