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| Mirrors > Home > MPE Home > Th. List > iotaeq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Andrew Salmon, 30-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| iotaeq | ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | drsb1 2500 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
| 2 | df-clab 2715 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
| 3 | df-clab 2715 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
| 4 | 1, 2, 3 | 3bitr4g 314 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})) |
| 5 | 4 | eqrdv 2735 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}) |
| 6 | 5 | eqeq1d 2739 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
| 7 | 6 | abbidv 2808 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}) |
| 8 | 7 | unieqd 4920 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}) |
| 9 | df-iota 6514 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
| 10 | df-iota 6514 | . 2 ⊢ (℩𝑦𝜑) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}} | |
| 11 | 8, 9, 10 | 3eqtr4g 2802 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 {csn 4626 ∪ cuni 4907 ℩cio 6512 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-13 2377 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 df-iota 6514 |
| This theorem is referenced by: (None) |
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