Theorem rexxp 4423

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 4343	. . 3 ⊢ ∪ y ∈ A ({y} × B) = (A × B)
2	1	rexeqi 2504	. 2 ⊢ (∃x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∃x ∈ (A × B)φ)
3		ralxp.1	. . 3 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
4	3	rexiunxp 4421	. 2 ⊢ (∃x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∃y ∈ A ∃z ∈ B ψ)
5	2, 4	bitr3i 175	1 ⊢ (∃x ∈ (A × B)φ ↔ ∃y ∈ A ∃z ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242  ∃wrex 2301  {csn 3367  ⟨cop 3370  ∪ ciun 3648   × cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by:  rexxpf  4426  fnrnov  5588  foov  5589  ovelimab  5593  cnref1o  8357