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Mirrors > Home > MPE Home > Th. List > spesbc | Structured version Visualization version GIF version |
Description: Existence form of spsbc 3804. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3801 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | rspesbca 3890 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
3 | 1, 2 | mpancom 688 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
4 | rexv 3507 | . 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 1776 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 [wsbc 3791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-sbc 3792 |
This theorem is referenced by: spesbcd 3892 opelopabsb 5540 sbccomieg 42781 frege124d 43751 sbiota1 44430 |
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