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| Description: form of spsbc 3801. (Contributed by Mario Carneiro, 9-Feb-2017.) | 
| Ref | Expression | 
|---|---|
| spesbcd.1 | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | 
| Ref | Expression | 
|---|---|
| spesbcd | ⊢ (𝜑 → ∃𝑥𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | spesbcd.1 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | |
| 2 | spesbc 3882 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∃𝑥𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wex 1779 [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: euotd 5518 ex-natded9.26 30438 bnj1465 34859 spesbcdi 38127 iscard4 43546 minregex 43547 brtrclfv2 43740 cotrclrcl 43755 rspesbcd 44958 | 
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