Theorem sbciegft 2869

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2870.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbciegft	⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbciegft

Step	Hyp	Ref	Expression
1		sbc5 2863	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
2		bi1 116	. . . . . . . 8 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	2	imim2i 12	. . . . . . 7 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜑 → 𝜓)))
4	3	impd 251	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ((𝑥 = 𝐴 ∧ 𝜑) → 𝜓))
5	4	alimi 1389	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓))
6		19.23t 1612	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓)))
7	6	biimpa 290	. . . . 5 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥((𝑥 = 𝐴 ∧ 𝜑) → 𝜓)) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓))
8	5, 7	sylan2 280	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓))
9	8	3adant1 961	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → 𝜓))
10	1, 9	syl5bi 150	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 → 𝜓))
11		bi2 128	. . . . . . . 8 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
12	11	imim2i 12	. . . . . . 7 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝑥 = 𝐴 → (𝜓 → 𝜑)))
13	12	com23 77	. . . . . 6 ⊢ ((𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝜓 → (𝑥 = 𝐴 → 𝜑)))
14	13	alimi 1389	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)))
15		19.21t 1519	. . . . . 6 ⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑)) ↔ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))))
16	15	biimpa 290	. . . . 5 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝜓 → (𝑥 = 𝐴 → 𝜑))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
17	14, 16	sylan2 280	. . . 4 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
18	17	3adant1 961	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑)))
19		sbc6g 2864	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
20	19	3ad2ant1 964	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
21	18, 20	sylibrd 167	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝜓 → [𝐴 / 𝑥]𝜑))
22	10, 21	impbid 127	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 924  ∀wal 1287   = wceq 1289  Ⅎwnf 1394  ∃wex 1426   ∈ wcel 1438  [wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841

This theorem is referenced by:  sbciegf  2870  sbciedf  2874