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Mirrors > Home > ILE Home > Th. List > opelopabf | GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4188 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
opelopabf.x | ⊢ Ⅎ𝑥𝜓 |
opelopabf.y | ⊢ Ⅎ𝑦𝜒 |
opelopabf.1 | ⊢ 𝐴 ∈ V |
opelopabf.2 | ⊢ 𝐵 ∈ V |
opelopabf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopabf.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opelopabf | ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabsb 4177 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
2 | opelopabf.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | nfcv 2279 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
4 | opelopabf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfsbc 2924 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 |
6 | opelopabf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | sbcbidv 2962 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
8 | 5, 7 | sbciegf 2935 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
9 | 2, 8 | ax-mp 5 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓) |
10 | opelopabf.2 | . . 3 ⊢ 𝐵 ∈ V | |
11 | opelopabf.y | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
12 | opelopabf.4 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
13 | 11, 12 | sbciegf 2935 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
14 | 10, 13 | ax-mp 5 | . 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒) |
15 | 1, 9, 14 | 3bitri 205 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 Vcvv 2681 [wsbc 2904 〈cop 3525 {copab 3983 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 |
This theorem is referenced by: pofun 4229 fmptco 5579 |
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