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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 5186 | . 2 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓) |
3 | rexv 3193 | . 2 ⊢ (∃𝑦 ∈ V ∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧 ∈ V 𝜓) | |
4 | rexv 3193 | . . 3 ⊢ (∃𝑧 ∈ V 𝜓 ↔ ∃𝑧𝜓) | |
5 | 4 | exbii 1764 | . 2 ⊢ (∃𝑦∃𝑧 ∈ V 𝜓 ↔ ∃𝑦∃𝑧𝜓) |
6 | 2, 3, 5 | 3bitri 285 | 1 ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∃wex 1695 ∃wrex 2897 Vcvv 3173 〈cop 4131 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-iun 4457 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: exopxfr2 5188 |
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