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Theorem bj-ssbid2ALT 33143
Description: Alternate proof of bj-ssbid2 33142, not using bj-ssbequ2 33140. (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 33118 . 2 ([𝑥/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 sp 2217 . . . . 5 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
32imim2i 16 . . . 4 ((𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
43alimi 1907 . . 3 (∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)))
5 pm2.21 121 . . . . . 6 𝑦 = 𝑥 → (𝑦 = 𝑥𝜑))
6 equcomi 2116 . . . . . . 7 (𝑦 = 𝑥𝑥 = 𝑦)
76imim1i 63 . . . . . 6 ((𝑥 = 𝑦𝜑) → (𝑦 = 𝑥𝜑))
85, 7ja 175 . . . . 5 ((𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → (𝑦 = 𝑥𝜑))
98alimi 1907 . . . 4 (∀𝑦(𝑦 = 𝑥 → (𝑥 = 𝑦𝜑)) → ∀𝑦(𝑦 = 𝑥𝜑))
10 ax6ev 2074 . . . 4 𝑦 𝑦 = 𝑥
11 19.23v 2038 . . . . 5 (∀𝑦(𝑦 = 𝑥𝜑) ↔ (∃𝑦 𝑦 = 𝑥𝜑))
1211biimpi 208 . . . 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 209 1 ([𝑥/𝑥]b𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wex 1875  [wssb 33117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-ssb 33118
This theorem is referenced by: (None)
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