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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 46976 | . . . . 5 ⊢ ∃!𝑧{𝑧} = {𝑦} | |
2 | eqcom 2742 | . . . . . 6 ⊢ ({𝑦} = {𝑧} ↔ {𝑧} = {𝑦}) | |
3 | 2 | eubii 2583 | . . . . 5 ⊢ (∃!𝑧{𝑦} = {𝑧} ↔ ∃!𝑧{𝑧} = {𝑦}) |
4 | 1, 3 | mpbir 231 | . . . 4 ⊢ ∃!𝑧{𝑦} = {𝑧} |
5 | eqeq1 2739 | . . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦} = {𝑧})) | |
6 | 5 | eubidv 2584 | . . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ ∃!𝑧{𝑦} = {𝑧})) |
7 | 4, 6 | mpbiri 258 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∃!𝑧{𝑥 ∣ 𝜑} = {𝑧}) |
8 | absn 4650 | . . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) | |
9 | reuabaiotaiota 47037 | . . . 4 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑)) | |
10 | eqcom 2742 | . . . 4 ⊢ ((℩𝑥𝜑) = (℩'𝑥𝜑) ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) | |
11 | 9, 10 | bitri 275 | . . 3 ⊢ (∃!𝑧{𝑥 ∣ 𝜑} = {𝑧} ↔ (℩'𝑥𝜑) = (℩𝑥𝜑)) |
12 | 7, 8, 11 | 3imtr3i 291 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = (℩𝑥𝜑)) |
13 | iotaval 6534 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦) | |
14 | 12, 13 | eqtrd 2775 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩'𝑥𝜑) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 ∃!weu 2566 {cab 2712 {csn 4631 ℩cio 6514 ℩'caiota 47033 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-sn 4632 df-pr 4634 df-uni 4913 df-int 4952 df-iota 6516 df-aiota 47035 |
This theorem is referenced by: aiota0def 47046 |
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