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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.13a | Structured version Visualization version GIF version |
Description: One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.13a | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 3413 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | biimpac 502 | 1 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-sbc 3403 |
This theorem is referenced by: pm13.194 37635 |
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