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Theorem sbcof2 1707
Description: Version of sbco 1858 where 𝑥 is not free in 𝜑. (Contributed by Jim Kingdon, 28-Dec-2017.)
Hypothesis
Ref Expression
sbcof2.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
sbcof2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbcof2
StepHypRef Expression
1 sbcof2.1 . . . . . . 7 (𝜑 → ∀𝑥𝜑)
21hbsb3 1705 . . . . . 6 ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
32sb6f 1700 . . . . 5 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
41sb6f 1700 . . . . . . 7 ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥𝜑))
54imbi2i 219 . . . . . 6 ((𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
65albii 1375 . . . . 5 (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
73, 6bitri 177 . . . 4 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
8 ax-11 1413 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑥(𝑥 = 𝑦 → (𝑦 = 𝑥𝜑))))
9 equcomi 1608 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
109imim1i 58 . . . . . . . . . 10 ((𝑦 = 𝑥𝜑) → (𝑥 = 𝑦𝜑))
1110imim2i 12 . . . . . . . . 9 ((𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
1211pm2.43d 48 . . . . . . . 8 ((𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦𝜑))
1312alimi 1360 . . . . . . 7 (∀𝑥(𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → ∀𝑥(𝑥 = 𝑦𝜑))
148, 13syl6 33 . . . . . 6 (𝑥 = 𝑦 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
1514a2i 11 . . . . 5 ((𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
1615alimi 1360 . . . 4 (∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)) → ∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
177, 16sylbi 118 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
18 ax-i9 1439 . . . . 5 𝑥 𝑥 = 𝑦
19 exim 1506 . . . . 5 (∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝑥(𝑥 = 𝑦𝜑)))
2018, 19mpi 15 . . . 4 (∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∃𝑥𝑥(𝑥 = 𝑦𝜑))
21 ax-ial 1443 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
222119.9h 1550 . . . 4 (∃𝑥𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2320, 22sylib 131 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥(𝑥 = 𝑦𝜑))
24 sb2 1666 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
2517, 23, 243syl 17 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 → [𝑦 / 𝑥]𝜑)
26 sb1 1665 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
27 simpl 106 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦)
28 19.8a 1498 . . . . . 6 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
2927, 28jca 294 . . . . 5 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
3029eximi 1507 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
319anim1i 327 . . . . . . . . 9 ((𝑥 = 𝑦𝜑) → (𝑦 = 𝑥𝜑))
3227, 31jca 294 . . . . . . . 8 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)))
3332eximi 1507 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)))
34 ax11e 1693 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)) → ∃𝑦(𝑦 = 𝑥𝜑)))
3533, 34syl5 32 . . . . . 6 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑦(𝑦 = 𝑥𝜑)))
3635imdistani 427 . . . . 5 ((𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → (𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
3736eximi 1507 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
3826, 30, 373syl 17 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
392sb5f 1701 . . . 4 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
401sb5f 1701 . . . . . 6 ([𝑥 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑥𝜑))
4140anbi2i 438 . . . . 5 ((𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4241exbii 1512 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4339, 42bitri 177 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4438, 43sylibr 141 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥][𝑥 / 𝑦]𝜑)
4525, 44impbii 121 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbid2h  1745
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