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Mirrors > Home > MPE Home > Th. List > Mathboxes > aiotaval | Structured version Visualization version GIF version |
Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of (alternate) iota. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
aiotaval | ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eusnsn 44192 | . . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦} | |
2 | eqcom 2744 | . . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦}) | |
3 | 2 | eubii 2584 | . . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦}) |
4 | 1, 3 | mpbir 234 | . . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧} |
5 | eqeq1 2741 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧})) | |
6 | 5 | eubidv 2585 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧})) |
7 | 4, 6 | mpbiri 261 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∃!𝑧{𝑥 ∣ 𝜑} = {𝑧}) |
8 | absn 4559 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
9 | reuabaiotaiota 44251 | . . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
10 | eqcom 2744 | . . . 4 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) | |
11 | 9, 10 | bitri 278 | . . 3 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) |
12 | 7, 8, 11 | 3imtr3i 294 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑)) |
13 | iotaval 6354 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
14 | 12, 13 | eqtrd 2777 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∃!weu 2567 {cab 2714 {csn 4541 ℩cio 6336 ℩'caiota 44247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-sn 4542 df-pr 4544 df-uni 4820 df-int 4860 df-iota 6338 df-aiota 44249 |
This theorem is referenced by: aiota0def 44260 |
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