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Mirrors > Home > ILE Home > Th. List > opelopabaf | GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4026 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
opelopabaf.x | ⊢ Ⅎ𝑥𝜓 |
opelopabaf.y | ⊢ Ⅎ𝑦𝜓 |
opelopabaf.1 | ⊢ 𝐴 ∈ V |
opelopabaf.2 | ⊢ 𝐵 ∈ V |
opelopabaf.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
opelopabaf | ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabsb 4015 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
2 | opelopabaf.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | opelopabaf.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | opelopabaf.x | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | opelopabaf.y | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
6 | nfv 1461 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V | |
7 | opelopabaf.3 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
8 | 4, 5, 6, 7 | sbc2iegf 2884 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 2, 3, 8 | mp2an 416 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
10 | 1, 9 | bitri 182 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Vcvv 2601 [wsbc 2815 〈cop 3401 {copab 3838 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 |
This theorem is referenced by: (None) |
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