Theorem rexcom13 2554

Description: Swap 1st and 3rd 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 2553	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑)
2		rexcom 2553	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
3	2	rexbii 2401	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
4		rexcom 2553	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitri 205	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 104  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381

This theorem is referenced by:  rexrot4  2555