Theorem exopxfr 5798

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 5797	. 2 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3		rexv 3457	. 2 ⊢ (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧 ∈ V 𝜓)
4		rexv 3457	. . 3 ⊢ (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
5	4	exbii 1850	. 2 ⊢ (∃𝑦∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧𝜓)
6	2, 3, 5	3bitri 297	1 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781  ∃wrex 3061  Vcvv 3429  ⟨cop 4573   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4935  df-opab 5148  df-xp 5637  df-rel 5638

This theorem is referenced by:  exopxfr2  5799