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Mirrors > Home > MPE Home > Th. List > sbcreu | Structured version Visualization version GIF version |
Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcreu | ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3412 | . 2 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V) | |
2 | reurex 3137 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) | |
3 | sbcex 3412 | . . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
4 | 3 | rexlimivw 3011 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
5 | 2, 4 | syl 17 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
6 | dfsbcq2 3405 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑)) | |
7 | dfsbcq2 3405 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
8 | 7 | reubidv 3103 | . . 3 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
9 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
10 | nfs1v 2425 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
11 | 9, 10 | nfreu 3093 | . . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
12 | sbequ12 2097 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | 12 | reubidv 3103 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
14 | 11, 13 | sbie 2396 | . . 3 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
15 | 6, 8, 14 | vtoclbg 3240 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
16 | 1, 5, 15 | pm5.21nii 367 | 1 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 [wsb 1867 ∈ wcel 1977 ∃wrex 2897 ∃!wreu 2898 Vcvv 3173 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-v 3175 df-sbc 3403 |
This theorem is referenced by: (None) |
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