| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5547 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 5535 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
| 2 | opelopabf.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 4 | opelopabf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 5 | 3, 4 | nfsbcw 3810 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 |
| 6 | opelopabf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | sbcbidv 3845 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
| 8 | 5, 7 | sbciegf 3827 | . . 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 3827 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
| 14 | 10, 13 | ax-mp 5 | . 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒) |
| 15 | 1, 9, 14 | 3bitri 297 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 〈cop 4632 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 |
| This theorem is referenced by: pofun 5610 fmptco 7149 fmptcof2 32667 |
| Copyright terms: Public domain | W3C validator |