Theorem bj-rexcom4 32063

Description: Remove from rexcom4 3198 dependency on ax-ext 2590 and ax-13 2234 (and on df-or 384, df-tru 1478, df-sb 1868, df-clab 2597, df-cleq 2603, df-clel 2606, df-nfc 2740, df-v 3175). This proof uses only df-rex 2902 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 2902	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
2		19.42v 1905	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑))
3	2	bicomi 213	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
4	3	exbii 1764	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑))
5		excom 2029	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rex 2902	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
7	6	bicomi 213	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)
8	7	exbii 1764	. . . 4 ⊢ (∃𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
9	5, 8	bitri 263	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
10	4, 9	bitri 263	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)
11	1, 10	bitri 263	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-11 2021

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-rex 2902

This theorem is referenced by:  bj-rexcom4a  32064