Theorem stdpc4lem 2091

Description: In the case of stdpc4 2092, rename-sb 2083 is derivable from fewer axioms than dfsb 2087. The essential proof step is presented in this lemma. Based on a proof of BJ, 22-Dec-2020. (Contributed by Wolf Lammen, 4-Jun-2026.)

Assertion

Ref	Expression
stdpc4lem	⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable groups:   𝑥,𝑦   𝑦,𝑡   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem stdpc4lem

Step	Hyp	Ref	Expression
1		ala1 1827	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 25	. 2 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1941	1 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-gen 1809  ax-4 1823  ax-5 1924

This theorem is referenced by:  stdpc4  2092