Theorem bj-rexcom4a 33097

Description: Remove from rexcom4a 3330 dependency on ax-ext 2704 and ax-13 2355 (and on df-or 384, df-sb 2011, df-clab 2711, df-cleq 2717, df-clel 2720, df-nfc 2855, df-v 3306). This proof uses only df-rex 3020 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 33096	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
2		19.42v 1994	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
3	2	rexbii 3143	. 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 383  ∃wex 1817  ∃wrex 3015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-11 2147

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1818  df-rex 3020

This theorem is referenced by:  bj-rexcom4bv  33098  bj-rexcom4b  33099