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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 4193 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 4182 | . 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 1508 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V | |
7 | opelopabaf.3 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
8 | 4, 5, 6, 7 | sbc2iegf 2979 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 2, 3, 8 | mp2an 422 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
10 | 1, 9 | bitri 183 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 Vcvv 2686 [wsbc 2909 〈cop 3530 {copab 3988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 |
This theorem is referenced by: (None) |
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