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Mirrors > Home > MPE Home > Th. List > opelopabaf | Structured version Visualization version GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5421 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 5409 | . 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 1911 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V | |
7 | opelopabaf.3 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
8 | 4, 5, 6, 7 | sbc2iegf 3848 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 2, 3, 8 | mp2an 690 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
10 | 1, 9 | bitri 277 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Vcvv 3494 [wsbc 3771 〈cop 4566 {copab 5120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 |
This theorem is referenced by: (None) |
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