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Mirrors > Home > MPE Home > Th. List > opelopabgf | Structured version Visualization version GIF version |
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5220 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.) |
Ref | Expression |
---|---|
opelopabgf.x | ⊢ Ⅎ𝑥𝜓 |
opelopabgf.y | ⊢ Ⅎ𝑦𝜒 |
opelopabgf.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopabgf.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opelopabgf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabsb 5212 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
2 | nfcv 2970 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
3 | opelopabgf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfsbc 3685 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 |
5 | opelopabgf.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 5 | sbcbidv 3718 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
7 | 4, 6 | sbciegf 3695 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) |
8 | opelopabgf.y | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
9 | opelopabgf.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | sbciegf 3695 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) |
11 | 7, 10 | sylan9bb 507 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜒)) |
12 | 1, 11 | syl5bb 275 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 [wsbc 3663 〈cop 4404 {copab 4936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4937 |
This theorem is referenced by: oprabv 6964 |
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