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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 2818 | . 2 ⊢ 𝑤 = 𝑤 | |
2 | oprabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓)) |
4 | 3 | oprabbidv 7209 | . 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 207 = wceq 1528 {coprab 7146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-oprab 7149 |
This theorem is referenced by: oprab4 7229 mpov 7253 dfxp3 7748 tposmpo 7918 addsrpr 10485 mulsrpr 10486 addcnsr 10545 mulcnsr 10546 joinfval2 17600 meetfval2 17614 dfxrn2 35508 |
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