Theorem rexcom13 2673

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 2671	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑)
2		rexcom 2671	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
3	2	rexbii 2514	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑)
4		rexcom 2671	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitri 206	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑧 ∈ 𝐶 ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 105  ∃wrex 2486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491

This theorem is referenced by:  rexrot4  2674