Theorem exopxfr 5831

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 5830	. 2 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓)
3		rexv 3482	. 2 ⊢ (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧 ∈ V 𝜓)
4		rexv 3482	. . 3 ⊢ (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓)
5	4	exbii 1878	. 2 ⊢ (∃𝑦∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧𝜓)
6	2, 3, 5	3bitri 300	1 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1809  ∃wrex 3089  Vcvv 3455  ⟨cop 4596   × cxp 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4959  df-opab 5175  df-xp 5669  df-rel 5670

This theorem is referenced by:  exopxfr2  5832