Theorem bj-ssbid2ALT 34771

Description: Alternate proof of bj-ssbid2 34770, not using sbequ2 2244. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ssbid2ALT	⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)

Proof of Theorem bj-ssbid2ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sb 2069	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sp 2178	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
3	2	imim2i 16	. . . 4 ⊢ ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑥 → (𝑥 = 𝑦 → 𝜑)))
4	3	alimi 1815	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦 → 𝜑)))
5		pm2.21 123	. . . . . 6 ⊢ (¬ 𝑦 = 𝑥 → (𝑦 = 𝑥 → 𝜑))
6		equcomi 2021	. . . . . . 7 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
7	6	imim1i 63	. . . . . 6 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑦 = 𝑥 → 𝜑))
8	5, 7	ja 186	. . . . 5 ⊢ ((𝑦 = 𝑥 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑥 → 𝜑))
9	8	alimi 1815	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑥 → 𝜑))
10		ax6ev 1974	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑥
11		19.23v 1946	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑥 → 𝜑))
12	11	biimpi 215	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) → (∃𝑦 𝑦 = 𝑥 → 𝜑))
13	9, 10, 12	mpisyl 21	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦 → 𝜑)) → 𝜑)
14	4, 13	syl 17	. 2 ⊢ (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → 𝜑)
15	1, 14	sylbi 216	1 ⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069

This theorem is referenced by: (None)