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Mirrors > Home > MPE Home > Th. List > spesbc | Structured version Visualization version GIF version |
Description: Existence form of spsbc 3776. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 3773 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | rspesbca 3853 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
3 | 1, 2 | mpancom 686 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
4 | rexv 3512 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
5 | 3, 4 | sylib 220 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 Vcvv 3486 [wsbc 3763 |
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 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3488 df-sbc 3764 |
This theorem is referenced by: spesbcd 3855 opelopabsb 5403 sbccomieg 39482 frege124d 40196 sbiota1 40856 |
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