Theorem rexcom13 3285

Description: Swap first and third restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)

Assertion

Ref	Expression
rexcom13	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑧,𝐵   𝑥,𝑦,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑧)

Proof of Theorem rexcom13

Step	Hyp	Ref	Expression
1		rexcom 3281	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑)
2		rexcom 3281	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
3	2	rexbii 3177	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
4		rexcom 3281	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitri 296	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 205  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-11 2156

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-rex 3069

This theorem is referenced by:  rexrot4  3287