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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.13b | Structured version Visualization version GIF version |
Description: Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.13b | ⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 3446 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | biimparc 504 | 1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-sbc 3436 |
This theorem is referenced by: pm14.24 38633 |
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