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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 5394 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 5382 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
2 | opelopabf.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
4 | opelopabf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfsbcw 3742 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 |
6 | opelopabf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | sbcbidv 3774 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
8 | 5, 7 | sbciegf 3757 | . . 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 3757 | . . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
14 | 10, 13 | ax-mp 5 | . 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒) |
15 | 1, 9, 14 | 3bitri 300 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 [wsbc 3720 〈cop 4531 {copab 5092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 |
This theorem is referenced by: pofun 5455 fmptco 6868 fmptcof2 30420 |
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