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Mirrors > Home > ILE Home > Th. List > sbcralg | GIF version |
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
sbcralg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq2 2965 | . 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑)) | |
2 | dfsbcq2 2965 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
3 | 2 | ralbidv 2477 | . 2 ⊢ (𝑧 = 𝐴 → (∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
4 | nfcv 2319 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | nfs1v 1939 | . . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
6 | 4, 5 | nfralxy 2515 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
7 | sbequ12 1771 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 7 | ralbidv 2477 | . . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
9 | 6, 8 | sbie 1791 | . 2 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
10 | 1, 3, 9 | vtoclbg 2798 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 [wsb 1762 ∈ wcel 2148 ∀wral 2455 [wsbc 2962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-sbc 2963 |
This theorem is referenced by: r19.12sn 3657 |
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