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Theorem bj-ssbid2ALT 31641
Description: Alternate proof of bj-ssbid2 31640, not using bj-ssbequ2 31638. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ssbid2ALT ([𝑥/𝑥]b𝜑𝜑)

Proof of Theorem bj-ssbid2ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-ssb 31615 . 2 ([𝑥/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sp 2040 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
32imim2i 16 . . . 4 ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
43alimi 1729 . . 3 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
5 pm2.21 118 . . . . . 6 𝑦 = 𝑥 → (𝑦 = 𝑥𝜑))
6 equcomi 1930 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
76imim1i 60 . . . . . 6 ((𝑥 = 𝑦𝜑) → (𝑦 = 𝑥𝜑))
85, 7ja 171 . . . . 5 ((𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥𝜑))
98alimi 1729 . . . 4 (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥𝜑))
10 ax6ev 1876 . . . 4 𝑦 𝑦 = 𝑥
11 19.23v 1888 . . . . 5 (∀𝑦(𝑦 = 𝑥𝜑) ↔ (∃𝑦 𝑦 = 𝑥𝜑))
1211biimpi 204 . . . 4 (∀𝑦(𝑦 = 𝑥𝜑) → (∃𝑦 𝑦 = 𝑥𝜑))
139, 10, 12mpisyl 21 . . 3 (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → 𝜑)
144, 13syl 17 . 2 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → 𝜑)
151, 14sylbi 205 1 ([𝑥/𝑥]b𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wex 1694  [wssb 31614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-ssb 31615
This theorem is referenced by: (None)
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