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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 3754 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | biimpac 480 | 1 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 [wsbc 3743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3744 |
This theorem is referenced by: pm13.194 42784 |
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