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Mirrors > Home > MPE Home > Th. List > sbid | Structured version Visualization version GIF version |
Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) |
Ref | Expression |
---|---|
sbid | ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2094 | . 2 ⊢ 𝑥 = 𝑥 | |
2 | sbequ12r 2259 | . 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 [wsb 2046 |
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-12 2196 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1854 df-sb 2047 |
This theorem is referenced by: sbco 2549 sbidm 2551 sbal2 2598 abid 2748 sbceq1a 3587 sbcid 3593 frege58bid 38698 sbidd 42972 sbidd-misc 42973 |
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