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Mirrors > Home > ILE Home > Th. List > spesbc | GIF version |
Description: Existence form of spsbc 2775. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Ref | Expression |
---|---|
spesbc | ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcex 2772 | . . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V) | |
2 | rspesbca 2842 | . . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑) | |
3 | 1, 2 | mpancom 399 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑) |
4 | rexv 2572 | . 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
5 | 3, 4 | sylib 127 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 Vcvv 2557 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 |
This theorem is referenced by: spesbcd 2844 opelopabsb 3997 |
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