Theorem rexxp 4407

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 4327	. . 3 ⊢ ∪ y ∈ A ({y} × B) = (A × B)
2	1	rexeqi 2488	. 2 ⊢ (∃x ∈ ∪ y ∈ A ({y} × B)φ ↔ ∃x ∈ (A × B)φ)
3		ralxp.1	. . 3 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
4	3	rexiunxp 4405	. 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 1228  ∃wrex 2285  {csn 3350  ⟨cop 3353  ∪ ciun 3631   × cxp 4270

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-iun 3633  df-opab 3793  df-xp 4278  df-rel 4279

This theorem is referenced by:  rexxpf  4410  fnrnov  5569  foov  5570  ovelimab  5574