Theorem bj-ssbequ2 33178

Description: Note that ax-12 2220 is used only via sp 2224. See sbequ2 2069 and stdpc7 2133. (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 33156	. . 3 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sp 2224	. . . . . 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 1910	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
5		pm3.31 442	. . . . 5 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
6	5	alimi 1910	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
7		19.23v 2041	. . . . 5 ⊢ (∀𝑦((𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑))
8		equvinva 2136	. . . . . . 7 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦))
9		biid 253	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
10		equcom 2122	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑦 ↔ 𝑦 = 𝑡)
11	9, 10	anbi12ci 621	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) ↔ (𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
12	11	biimpi 208	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → (𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
13	12	eximi 1933	. . . . . . . . 9 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → ∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦))
14		pm3.35 837	. . . . . . . . 9 ⊢ ((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑)) → 𝜑)
15	13, 14	sylan 575	. . . . . . . 8 ⊢ ((∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) ∧ (∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑)) → 𝜑)
16	15	ancoms 452	. . . . . . 7 ⊢ (((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ∧ ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦)) → 𝜑)
17	8, 16	sylan2 586	. . . . . 6 ⊢ (((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) ∧ 𝑥 = 𝑡) → 𝜑)
18	17	ex 403	. . . . 5 ⊢ ((∃𝑦(𝑦 = 𝑡 ∧ 𝑥 = 𝑦) → 𝜑) → (𝑥 = 𝑡 → 𝜑))
19	7, 18	sylbi 209	. . . 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 209	. 2 ⊢ ([𝑡/𝑥]b𝜑 → (𝑥 = 𝑡 → 𝜑))
22	21	com12 32	1 ⊢ (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1654  ∃wex 1878  [wssb 33155

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-ssb 33156

This theorem is referenced by:  bj-ssbid2  33180