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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 3784 | . 2 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V) | |
2 | reurex 3433 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) | |
3 | sbcex 3784 | . . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
4 | 3 | rexlimivw 3284 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
5 | 2, 4 | syl 17 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) |
6 | dfsbcq2 3777 | . . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑)) | |
7 | dfsbcq2 3777 | . . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
8 | 7 | reubidv 3391 | . . 3 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
9 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
10 | nfs1v 2160 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
11 | 9, 10 | nfreuw 3376 | . . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 |
12 | sbequ12 2253 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
13 | 12 | reubidv 3391 | . . . 4 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)) |
14 | 11, 13 | sbiev 2330 | . . 3 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) |
15 | 6, 8, 14 | vtoclbg 3571 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)) |
16 | 1, 5, 15 | pm5.21nii 382 | 1 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 [wsb 2069 ∈ wcel 2114 ∃wrex 3141 ∃!wreu 3142 Vcvv 3496 [wsbc 3774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-v 3498 df-sbc 3775 |
This theorem is referenced by: (None) |
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