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Mirrors > Home > ILE Home > Th. List > opabbii | GIF version |
Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.) |
Ref | Expression |
---|---|
opabbii.1 | ⊢ (φ ↔ ψ) |
Ref | Expression |
---|---|
opabbii | ⊢ {〈x, y〉 ∣ φ} = {〈x, y〉 ∣ ψ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2037 | . 2 ⊢ z = z | |
2 | opabbii.1 | . . . 4 ⊢ (φ ↔ ψ) | |
3 | 2 | a1i 9 | . . 3 ⊢ (z = z → (φ ↔ ψ)) |
4 | 3 | opabbidv 3814 | . 2 ⊢ (z = z → {〈x, y〉 ∣ φ} = {〈x, y〉 ∣ ψ}) |
5 | 1, 4 | ax-mp 7 | 1 ⊢ {〈x, y〉 ∣ φ} = {〈x, y〉 ∣ ψ} |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1242 {copab 3808 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-opab 3810 |
This theorem is referenced by: mptv 3844 fconstmpt 4330 xpundi 4339 xpundir 4340 inxp 4413 cnvco 4463 resopab 4595 opabresid 4602 cnvi 4671 cnvun 4672 cnvin 4674 cnvxp 4685 cnvcnv3 4713 coundi 4765 coundir 4766 mptun 4972 fvopab6 5207 cbvoprab1 5518 cbvoprab12 5520 dmoprabss 5528 mpt2mptx 5537 resoprab 5539 ov6g 5580 dfoprab3s 5758 dfoprab3 5759 dfoprab4 5760 xpcomco 6236 dmaddpq 6363 dmmulpq 6364 recmulnqg 6375 enq0enq 6414 ltrelxr 6877 ltxr 8465 |
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