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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 3587 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | biimparc 505 | 1 ⊢ (([𝐴 / 𝑥]𝜑 ∧ 𝑥 = 𝐴) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 [wsbc 3576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-12 2196 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1854 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-sbc 3577 |
This theorem is referenced by: pm14.24 39153 |
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