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Mirrors > Home > MPE Home > Th. List > copsex2ga | Structured version Visualization version GIF version |
Description: Implicit substitution inference for ordered pairs. Compare copsex2g 5387. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
copsex2ga.1 | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
copsex2ga | ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss 5574 | . . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V) | |
2 | 1 | sseli 3966 | . 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V)) |
3 | copsex2ga.1 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
4 | 3 | copsex2gb 5682 | . . 3 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑)) |
5 | 4 | baibr 539 | . 2 ⊢ (𝐴 ∈ (V × V) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 ∈ wcel 2113 Vcvv 3497 〈cop 4576 × cxp 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-opab 5132 df-xp 5564 |
This theorem is referenced by: (None) |
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