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| Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5543 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 5535 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑) | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 3 | opelopabgf.x | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
| 4 | 2, 3 | nfsbcw 3810 | . . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓 | 
| 5 | opelopabgf.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 6 | 5 | sbcbidv 3845 | . . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)) | 
| 7 | 4, 6 | sbciegf 3827 | . . 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 3827 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒)) | 
| 11 | 7, 10 | sylan9bb 509 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜒)) | 
| 12 | 1, 11 | bitrid 283 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 [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: oprabv 7493 | 
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