Theorem rexcom4a 3270

Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Assertion

Ref	Expression
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 rexcom4a

Step	Hyp	Ref	Expression
1		rexcom4 3267	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
2		19.42v 1960	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
3	2	rexbii 3087	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))
4	1, 3	bitr3i 278	1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∃wex 1786  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-11 2168

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-rex 3065

This theorem is referenced by:  rexcom4b  3464  bj-rexcom4bv  37242  bj-rexcom4b  37243  tfsconcatlem  43788