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| Mirrors > Home > MPE Home > Th. List > oprabbii | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal operation class abstractions. (Contributed by NM, 28-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
| Ref | Expression |
|---|---|
| oprabbii.1 | ⊢ (𝜑 ↔ 𝜓) |
| Ref | Expression |
|---|---|
| oprabbii | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ 𝑤 = 𝑤 | |
| 2 | oprabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓)) |
| 4 | 3 | oprabbidv 7499 | . 2 ⊢ (𝑤 = 𝑤 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓}) |
| 5 | 1, 4 | ax-mp 5 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 {coprab 7432 |
| 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-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-oprab 7435 |
| This theorem is referenced by: oprab4 7519 mpov 7545 dfxp3 8086 tposmpo 8288 addsrpr 11115 mulsrpr 11116 addcnsr 11175 mulcnsr 11176 joinfval2 18419 meetfval2 18433 dfxrn2 38377 |
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