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Mirrors > Home > MPE Home > Th. List > opelopabf | Structured version Visualization version GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4922 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 4910 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
2 | opelopabf.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
4 | opelopabf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfsbc 3424 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 |
6 | opelopabf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | sbcbidv 3457 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
8 | 5, 7 | sbciegf 3434 | . . 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 3434 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
14 | 10, 13 | ax-mp 5 | . 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒) |
15 | 1, 9, 14 | 3bitri 285 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Vcvv 3173 [wsbc 3402 〈cop 4131 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 |
This theorem is referenced by: pofun 4975 fmptco 6303 fmptcof2 28839 |
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