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Mirrors > Home > MPE Home > Th. List > exopxfr | Structured version Visualization version GIF version |
Description: Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
exopxfr.1 | ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
exopxfr | ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exopxfr.1 | . . 3 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓)) | |
2 | 1 | rexxp 5803 | . 2 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓) |
3 | rexv 3473 | . 2 ⊢ (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧 ∈ V 𝜓) | |
4 | rexv 3473 | . . 3 ⊢ (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓) | |
5 | 4 | exbii 1851 | . 2 ⊢ (∃𝑦∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧𝜓) |
6 | 2, 3, 5 | 3bitri 297 | 1 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∃wrex 3074 Vcvv 3448 ⟨cop 4597 × cxp 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-iun 4961 df-opab 5173 df-xp 5644 df-rel 5645 |
This theorem is referenced by: exopxfr2 5805 |
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