Theorem bj-ssbid1ALT 34773

Description: Alternate proof of bj-ssbid1 34772, not using sbequ1 2243. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bj-ssbid1ALT	⊢ (𝜑 → [𝑥 / 𝑥]𝜑)

Proof of Theorem bj-ssbid1ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax12v 2174	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	equcoms 2024	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	com12 32	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	alrimiv 1931	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5		df-sb 2069	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	4, 5	sylibr 233	1 ⊢ (𝜑 → [𝑥 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537  [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)