Theorem stdpc4ALT 2093

Description: Alternate proof of stdpc4 2092, shorter but using additional axioms. (Contributed by WL, 5-Jun-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
stdpc4ALT	⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Proof of Theorem stdpc4ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ala1 1827	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 25	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1941	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		dfsb 2087	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	3, 4	sylibr 236	1 ⊢ (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1552  [wsb 2084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1794  df-sb 2085

This theorem is referenced by: (None)