Theorem rexxp 4683

Description: Existential quantification restricted to a cross product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)

Hypothesis

Ref	Expression
ralxp.1	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexxp	⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥   𝑦,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem rexxp

Step	Hyp	Ref	Expression
1		iunxpconst 4599	. . 3 ⊢ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵) = (𝐴 × 𝐵)
2	1	rexeqi 2631	. 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝜑)
3		ralxp.1	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
4	3	rexiunxp 4681	. 2 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
5	2, 4	bitr3i 185	1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃wrex 2417  {csn 3527  ⟨cop 3530  ∪ ciun 3813   × cxp 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-opab 3990  df-xp 4545  df-rel 4546

This theorem is referenced by:  rexxpf  4686  fnrnov  5916  foov  5917  ovelimab  5921  xpf1o  6738  cnref1o  9443  txbas  12430