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| Mirrors > Home > MPE Home > Th. List > sbcreu | Structured version Visualization version GIF version | ||
| Description: Interchange class substitution and restricted unique existential 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 3798 | . 2 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V) | |
| 2 | reurex 3384 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) | |
| 3 | sbcex 3798 | . . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
| 4 | 3 | rexlimivw 3151 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
| 5 | 2, 4 | syl 17 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
| 6 | dfsbcq2 3791 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑)) | |
| 7 | dfsbcq2 3791 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 8 | 7 | reubidv 3398 | . . 3 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
| 9 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 10 | nfs1v 2156 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
| 11 | 9, 10 | nfreuw 3414 | . . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
| 12 | sbequ12 2251 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
| 13 | 12 | reubidv 3398 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
| 14 | 11, 13 | sbiev 2314 | . . 3 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
| 15 | 6, 8, 14 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
| 16 | 1, 5, 15 | pm5.21nii 378 | 1 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 [wsb 2064 ∈ wcel 2108 ∃wrex 3070 ∃!wreu 3378 Vcvv 3480 [wsbc 3788 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-rmo 3380 df-reu 3381 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: (None) |
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