Theorem bj-rexcom4a 32870

Description: Remove from rexcom4a 3226 dependency on ax-ext 2602 and ax-13 2246 (and on df-or 385, 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, 16-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-rexcom4a	⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem bj-rexcom4a

Step	Hyp	Ref	Expression
1		bj-rexcom4 32869	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
2		19.42v 1918	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
3	2	rexbii 3041	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))
4	1, 3	bitr3i 266	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∃wex 1704  ∃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-rexcom4bv  32871  bj-rexcom4b  32872