Theorem rexcom 2545

Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
rexcom	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

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

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

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		nfcv 2235	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2235	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	rexcomf 2543	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 104  ∃wrex 2371

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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376

This theorem is referenced by:  rexcom13  2546  rexcom4  2656  iuncom  3758  xpiundi  4525  addcomprg  7234  mulcomprg  7236  ltexprlemm  7256  caucvgprprlemexbt  7362  qmulz  9207  caubnd2  10681  sqrt2irr  11583