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Theorem bj-ssbid1ALT 32773
 Description: Alternate proof of bj-ssbid1 32772, not using bj-ssbequ1 32769. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-ssbid1ALT (𝜑 → [𝑥/𝑥]b𝜑)

Proof of Theorem bj-ssbid1ALT
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ax12v 2088 . . . . 5 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
21equcoms 1993 . . . 4 (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
32com12 32 . . 3 (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
43alrimiv 1895 . 2 (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
5 df-ssb 32745 . 2 ([𝑥/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦𝜑)))
64, 5sylibr 224 1 (𝜑 → [𝑥/𝑥]b𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1521  [wssb 32744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-ssb 32745 This theorem is referenced by: (None)
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