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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 4997 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 4985 | . 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 1843 | . . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V | |
7 | opelopabaf.3 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
8 | 4, 5, 6, 7 | sbc2iegf 3504 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)) |
9 | 2, 3, 8 | mp2an 708 | . 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓) |
10 | 1, 9 | bitri 264 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Vcvv 3200 [wsbc 3435 〈cop 4183 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 |
This theorem is referenced by: (None) |
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