Theorem bj-ssbid1ALT 37328

Description: Alternate proof of bj-ssbid1 37327, not using sbequ1 2287. (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 2217	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	equcoms 2053	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	com12 33	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	alrimiv 1960	. 2 ⊢ (𝜑 → ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5		dfsb 2101	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	4, 5	sylibr 237	1 ⊢ (𝜑 → [𝑥 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by: (None)