Theorem exopxfr 5834

Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
exopxfr	⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)

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

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

Proof of Theorem exopxfr

Step	Hyp	Ref	Expression
1		exopxfr.1	. . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
2	1	rexxp 5833	. 2 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3		rexv 3485	. 2 ⊢ (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧 ∈ V 𝜓)
4		rexv 3485	. . 3 ⊢ (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
5	4	exbii 1881	. 2 ⊢ (∃𝑦∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧𝜓)
6	2, 3, 5	3bitri 300	1 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812  ∃wrex 3092  Vcvv 3458  ⟨cop 4600   × cxp 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-iun 4963  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by:  exopxfr2  5835