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Theorem sbcof2 1691
Description: Version of sbco 1842 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 1689 . . . . . 6 ([𝑥 / 𝑦]𝜑 → ∀𝑦[𝑥 / 𝑦]𝜑)
32sb6f 1684 . . . . 5 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑))
41sb6f 1684 . . . . . . 7 ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥𝜑))
54imbi2i 215 . . . . . 6 ((𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ (𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
65albii 1359 . . . . 5 (∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
73, 6bitri 173 . . . 4 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)))
8 ax-11 1397 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑥(𝑥 = 𝑦 → (𝑦 = 𝑥𝜑))))
9 equcomi 1592 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
109imim1i 54 . . . . . . . . . 10 ((𝑦 = 𝑥𝜑) → (𝑥 = 𝑦𝜑))
1110imim2i 12 . . . . . . . . 9 ((𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦 → (𝑥 = 𝑦𝜑)))
1211pm2.43d 44 . . . . . . . 8 ((𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦𝜑))
1312alimi 1344 . . . . . . 7 (∀𝑥(𝑥 = 𝑦 → (𝑦 = 𝑥𝜑)) → ∀𝑥(𝑥 = 𝑦𝜑))
148, 13syl6 29 . . . . . 6 (𝑥 = 𝑦 → (∀𝑦(𝑦 = 𝑥𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
1514a2i 11 . . . . 5 ((𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)) → (𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
1615alimi 1344 . . . 4 (∀𝑥(𝑥 = 𝑦 → ∀𝑦(𝑦 = 𝑥𝜑)) → ∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
177, 16sylbi 114 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)))
18 ax-i9 1423 . . . . 5 𝑥 𝑥 = 𝑦
19 exim 1490 . . . . 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 1427 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
222119.9h 1534 . . . 4 (∃𝑥𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
2320, 22sylib 127 . . 3 (∀𝑥(𝑥 = 𝑦 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑥(𝑥 = 𝑦𝜑))
24 sb2 1650 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
2517, 23, 243syl 17 . 2 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 → [𝑦 / 𝑥]𝜑)
26 sb1 1649 . . . 4 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
27 simpl 102 . . . . . 6 ((𝑥 = 𝑦𝜑) → 𝑥 = 𝑦)
28 19.8a 1482 . . . . . 6 ((𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
2927, 28jca 290 . . . . 5 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
3029eximi 1491 . . . 4 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
319anim1i 323 . . . . . . . . 9 ((𝑥 = 𝑦𝜑) → (𝑦 = 𝑥𝜑))
3227, 31jca 290 . . . . . . . 8 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)))
3332eximi 1491 . . . . . . 7 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)))
34 ax11e 1677 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ (𝑦 = 𝑥𝜑)) → ∃𝑦(𝑦 = 𝑥𝜑)))
3533, 34syl5 28 . . . . . 6 (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑦(𝑦 = 𝑥𝜑)))
3635imdistani 419 . . . . 5 ((𝑥 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → (𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
3736eximi 1491 . . . 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 1685 . . . 4 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
401sb5f 1685 . . . . . 6 ([𝑥 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑥𝜑))
4140anbi2i 430 . . . . 5 ((𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ (𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4241exbii 1496 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4339, 42bitri 173 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ ∃𝑦(𝑦 = 𝑥𝜑)))
4438, 43sylibr 137 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥][𝑥 / 𝑦]𝜑)
4525, 44impbii 117 1 ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wal 1241  wex 1381  [wsb 1645
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 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-sb 1646
This theorem is referenced by:  sbid2h  1729
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