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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 2725 | . 2 ⊢ 𝑤 = 𝑤 | |
2 | oprabbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑤 = 𝑤 → (𝜑 ↔ 𝜓)) |
4 | 3 | oprabbidv 7486 | . 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 205 = wceq 1533 {coprab 7420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-oprab 7423 |
This theorem is referenced by: oprab4 7506 mpov 7532 dfxp3 8066 tposmpo 8269 addsrpr 11100 mulsrpr 11101 addcnsr 11160 mulcnsr 11161 joinfval2 18369 meetfval2 18383 dfxrn2 37978 |
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