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Mirrors > Home > MPE Home > Th. List > copsex2gb | Structured version Visualization version GIF version |
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5673. (Contributed by NM, 12-Mar-2014.) |
Ref | Expression |
---|---|
copsex2ga.1 | ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
copsex2gb | ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 5619 | . . 3 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | 1 | anbi1i 625 | . 2 ⊢ ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) |
3 | 19.41vv 1945 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
4 | copsex2ga.1 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (𝜑 ↔ 𝜓)) | |
5 | 4 | pm5.32i 577 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
6 | 5 | 2exbii 1843 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
7 | 2, 3, 6 | 3bitr2ri 302 | 1 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 Vcvv 3493 〈cop 4565 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-opab 5120 df-xp 5554 |
This theorem is referenced by: copsex2ga 5673 elopaba 5674 dfxrn2 35620 elcnvlem 39952 |
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