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| Mirrors > Home > MPE Home > Th. List > equsb1 | Structured version Visualization version GIF version | ||
| Description: Substitution applied to an atomic wff. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker equsb1v 2105 if possible. (Contributed by NM, 10-May-1993.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| equsb1 | ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sb2 2484 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝑥 = 𝑦) → [𝑦 / 𝑥]𝑥 = 𝑦) | |
| 2 | id 22 | . 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦) | |
| 3 | 1, 2 | mpg 1797 | 1 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 [wsb 2064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 | 
| This theorem is referenced by: sbequ8 2506 sbie 2507 frege54cor1b 43907 sb5ALT 44545 sb5ALTVD 44933 | 
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