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| Mirrors > Home > MPE Home > Th. List > spesbc | Structured version Visualization version GIF version | ||
| Description: Existence form of spsbc 3801. (Contributed by Mario Carneiro, 18-Nov-2016.) |
| Ref | Expression |
|---|---|
| spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sbcex 3798 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
| 2 | rspesbca 3881 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
| 3 | 1, 2 | mpancom 688 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
| 4 | rexv 3509 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
| 5 | 3, 4 | sylib 218 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 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-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-sbc 3789 |
| This theorem is referenced by: spesbcd 3883 opelopabsb 5535 sbccomieg 42804 frege124d 43774 sbiota1 44453 |
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