Theorem bj-ssbequ2 32643

Description: Note that ax-12 2047 is used only via sp 2053. See sbequ2 1882 and stdpc7 1958. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbequ2	⊢ (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑 → 𝜑))

Proof of Theorem bj-ssbequ2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ssb 32620	. . 3 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sp 2053	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
3	2	imim2i 16	. . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
4	3	alimi 1739	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
5		pm3.31 461	. . . . 5 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
6	5	alimi 1739	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
7		19.23v 1902	. . . . 5 ⊢ (∀𝑦((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
8		equviniva 1960	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦))
9		biid 251	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
10		equcom 1945	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑦 ↔ 𝑦 = 𝑡)
11	9, 10	anbi12ci 734	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) ↔ (𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
12	11	biimpi 206	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → (𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
13	12	eximi 1762	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → ∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
14		pm3.35 611	. . . . . . . . 9 ⊢ ((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑)) → 𝜑)
15	13, 14	sylan 488	. . . . . . . 8 ⊢ ((∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑)) → 𝜑)
16	15	ancoms 469	. . . . . . 7 ⊢ (((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ∧ ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦)) → 𝜑)
17	8, 16	sylan2 491	. . . . . 6 ⊢ (((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ∧ 𝑥 = 𝑡) → 𝜑)
18	17	ex 450	. . . . 5 ⊢ ((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) → (𝑥 = 𝑡 → 𝜑))
19	7, 18	sylbi 207	. . . 4 ⊢ (∀𝑦((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) → (𝑥 = 𝑡 → 𝜑))
20	4, 6, 19	3syl 18	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑥 = 𝑡 → 𝜑))
21	1, 20	sylbi 207	. 2 ⊢ ([𝑡/𝑥]b𝜑 → (𝑥 = 𝑡 → 𝜑))
22	21	com12 32	1 ⊢ (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by:  bj-ssbid2  32645