Theorem sbtlem 2070

Description: In the case of sbt 2071, the hypothesis in df-sb 2068 is derivable from propositional axioms and ax-gen 1796 alone. The essential proof step is presented in this lemma. (Contributed by Wolf Lammen, 4-Feb-2026.)

Hypothesis

Ref	Expression
sbtlem.1	⊢ 𝜑

Assertion

Ref	Expression
sbtlem	⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem sbtlem

Step	Hyp	Ref	Expression
1		sbtlem.1	. . . . 5 ⊢ 𝜑
2	1	a1i 11	. . . 4 ⊢ (𝑥 = 𝑦 → 𝜑)
3	2	ax-gen 1796	. . 3 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)
4	3	a1i 11	. 2 ⊢ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	ax-gen 1796	1 ⊢ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-gen 1796

This theorem is referenced by:  sbt  2071