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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.195 | Structured version Visualization version GIF version |
Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3799. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
pm13.195 | ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc5 3799 | . 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑)) | |
2 | 1 | bicomi 226 | 1 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3772 |
This theorem is referenced by: (None) |
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