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| Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) | 
| Ref | Expression | 
|---|---|
| iotavalsb | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 19.8a 2180 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 2 | eu6 2573 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
| 3 | iotavalb 44454 | . . . 4 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (℩𝑥𝜑) = 𝑦)) | |
| 4 | dfsbcq 3789 | . . . . 5 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | |
| 5 | 4 | eqcoms 2744 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑦 → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | 
| 6 | 3, 5 | biimtrdi 253 | . . 3 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))) | 
| 7 | 2, 6 | sylbir 235 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓))) | 
| 8 | 1, 7 | mpcom 38 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([𝑦 / 𝑧]𝜓 ↔ [(℩𝑥𝜑) / 𝑧]𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∃wex 1778 ∃!weu 2567 [wsbc 3787 ℩cio 6511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-sbc 3788 df-un 3955 df-ss 3967 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: (None) | 
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