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Mirrors > Home > ILE Home > Th. List > sbcalg | GIF version |
Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Ref | Expression |
---|---|
sbcalg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2818 | . 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑)) | |
2 | dfsbcq2 2818 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑)) | |
3 | 2 | albidv 1745 | . 2 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
4 | sbal 1917 | . 2 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑) | |
5 | 1, 3, 4 | vtoclbg 2659 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 [wsb 1685 [wsbc 2815 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-sbc 2816 |
This theorem is referenced by: sbcabel 2895 sbcssg 3350 |
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