Metamath Proof Explorer


Theorem bj-ssbid2ALT

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

Ref Expression
Assertion bj-ssbid2ALT ( [ 𝑥 / 𝑥 ] 𝜑𝜑 )

Proof

Step Hyp Ref Expression
1 df-sb ( [ 𝑥 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )
2 sp ( ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) → ( 𝑥 = 𝑦𝜑 ) )
3 2 imim2i ( ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) → ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦𝜑 ) ) )
4 3 alimi ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) → ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦𝜑 ) ) )
5 pm2.21 ( ¬ 𝑦 = 𝑥 → ( 𝑦 = 𝑥𝜑 ) )
6 equcomi ( 𝑦 = 𝑥𝑥 = 𝑦 )
7 6 imim1i ( ( 𝑥 = 𝑦𝜑 ) → ( 𝑦 = 𝑥𝜑 ) )
8 5 7 ja ( ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦𝜑 ) ) → ( 𝑦 = 𝑥𝜑 ) )
9 8 alimi ( ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦𝜑 ) ) → ∀ 𝑦 ( 𝑦 = 𝑥𝜑 ) )
10 ax6ev 𝑦 𝑦 = 𝑥
11 19.23v ( ∀ 𝑦 ( 𝑦 = 𝑥𝜑 ) ↔ ( ∃ 𝑦 𝑦 = 𝑥𝜑 ) )
12 11 biimpi ( ∀ 𝑦 ( 𝑦 = 𝑥𝜑 ) → ( ∃ 𝑦 𝑦 = 𝑥𝜑 ) )
13 9 10 12 mpisyl ( ∀ 𝑦 ( 𝑦 = 𝑥 → ( 𝑥 = 𝑦𝜑 ) ) → 𝜑 )
14 4 13 syl ( ∀ 𝑦 ( 𝑦 = 𝑥 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) → 𝜑 )
15 1 14 sylbi ( [ 𝑥 / 𝑥 ] 𝜑𝜑 )