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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 2370. (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 2497 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
2 | df-clab 2714 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
3 | df-clab 2714 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
4 | 1, 2, 3 | 3bitr4g 313 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜑})) |
5 | 4 | eqrdv 2734 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜑}) |
6 | 5 | eqeq1d 2738 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑧} ↔ {𝑦 ∣ 𝜑} = {𝑧})) |
7 | 6 | abbidv 2805 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}) |
8 | 7 | unieqd 4866 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}}) |
9 | df-iota 6431 | . 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ {𝑥 ∣ 𝜑} = {𝑧}} | |
10 | df-iota 6431 | . 2 ⊢ (℩𝑦𝜑) = ∪ {𝑧 ∣ {𝑦 ∣ 𝜑} = {𝑧}} | |
11 | 8, 9, 10 | 3eqtr4g 2801 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (℩𝑥𝜑) = (℩𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 [wsb 2066 ∈ wcel 2105 {cab 2713 {csn 4573 ∪ cuni 4852 ℩cio 6429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-12 2170 ax-13 2370 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-in 3905 df-ss 3915 df-uni 4853 df-iota 6431 |
This theorem is referenced by: (None) |
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