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Theorem difopab 4392
Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
difopab ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ¬ ψ)}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem difopab
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4387 . . 3 Rel {⟨x, y⟩ ∣ φ}
2 reldif 4380 . . 3 (Rel {⟨x, y⟩ ∣ φ} → Rel ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}))
31, 2ax-mp 7 . 2 Rel ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ})
4 relopab 4387 . 2 Rel {⟨x, y⟩ ∣ (φ ¬ ψ)}
5 sbcan 2778 . . . 4 ([z / x]([w / y]φ [w / y] ¬ ψ) ↔ ([z / x][w / y]φ [z / x][w / y] ¬ ψ))
6 sbcan 2778 . . . . 5 ([w / y](φ ¬ ψ) ↔ ([w / y]φ [w / y] ¬ ψ))
76sbcbii 2791 . . . 4 ([z / x][w / y](φ ¬ ψ) ↔ [z / x]([w / y]φ [w / y] ¬ ψ))
8 opelopabsb 3967 . . . . 5 (⟨z, w {⟨x, y⟩ ∣ φ} ↔ [z / x][w / y]φ)
9 vex 2534 . . . . . . 7 z V
10 sbcng 2776 . . . . . . 7 (z V → ([z / x] ¬ [w / y]ψ ↔ ¬ [z / x][w / y]ψ))
119, 10ax-mp 7 . . . . . 6 ([z / x] ¬ [w / y]ψ ↔ ¬ [z / x][w / y]ψ)
12 vex 2534 . . . . . . . 8 w V
13 sbcng 2776 . . . . . . . 8 (w V → ([w / y] ¬ ψ ↔ ¬ [w / y]ψ))
1412, 13ax-mp 7 . . . . . . 7 ([w / y] ¬ ψ ↔ ¬ [w / y]ψ)
1514sbcbii 2791 . . . . . 6 ([z / x][w / y] ¬ ψ[z / x] ¬ [w / y]ψ)
16 opelopabsb 3967 . . . . . . 7 (⟨z, w {⟨x, y⟩ ∣ ψ} ↔ [z / x][w / y]ψ)
1716notbii 581 . . . . . 6 (¬ ⟨z, w {⟨x, y⟩ ∣ ψ} ↔ ¬ [z / x][w / y]ψ)
1811, 15, 173bitr4ri 202 . . . . 5 (¬ ⟨z, w {⟨x, y⟩ ∣ ψ} ↔ [z / x][w / y] ¬ ψ)
198, 18anbi12i 436 . . . 4 ((⟨z, w {⟨x, y⟩ ∣ φ} ¬ ⟨z, w {⟨x, y⟩ ∣ ψ}) ↔ ([z / x][w / y]φ [z / x][w / y] ¬ ψ))
205, 7, 193bitr4ri 202 . . 3 ((⟨z, w {⟨x, y⟩ ∣ φ} ¬ ⟨z, w {⟨x, y⟩ ∣ ψ}) ↔ [z / x][w / y](φ ¬ ψ))
21 eldif 2900 . . 3 (⟨z, w ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}) ↔ (⟨z, w {⟨x, y⟩ ∣ φ} ¬ ⟨z, w {⟨x, y⟩ ∣ ψ}))
22 opelopabsb 3967 . . 3 (⟨z, w {⟨x, y⟩ ∣ (φ ¬ ψ)} ↔ [z / x][w / y](φ ¬ ψ))
2320, 21, 223bitr4i 201 . 2 (⟨z, w ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}) ↔ ⟨z, w {⟨x, y⟩ ∣ (φ ¬ ψ)})
243, 4, 23eqrelriiv 4357 1 ({⟨x, y⟩ ∣ φ} ∖ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ¬ ψ)}
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97  wb 98   = wceq 1226   wcel 1370  Vcvv 2531  [wsbc 2737  cdif 2887  cop 3349  {copab 3787  Rel wrel 4273
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-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274  df-rel 4275
This theorem is referenced by: (None)
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