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Mirrors > Home > MPE Home > Th. List > spesbc | Structured version Visualization version GIF version |
Description: Existence form of spsbc 3787. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3784 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | rspesbca 3871 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
3 | 1, 2 | mpancom 687 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
4 | rexv 3495 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
5 | 3, 4 | sylib 217 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1774 ∈ wcel 2099 ∃wrex 3065 Vcvv 3469 [wsbc 3774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-v 3471 df-sbc 3775 |
This theorem is referenced by: spesbcd 3873 opelopabsb 5526 sbccomieg 42135 frege124d 43114 sbiota1 43794 |
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