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Mirrors > Home > MPE Home > Th. List > spesbc | Structured version Visualization version GIF version |
Description: Existence form of spsbc 3415. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3412 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | rspesbca 3486 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
3 | 1, 2 | mpancom 700 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
4 | rexv 3193 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
5 | 3, 4 | sylib 207 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 |
This theorem is referenced by: spesbcd 3488 opelopabsb 4910 sbccomieg 36375 frege124d 37072 sbiota1 37657 |
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