Theorem bj-alsb 32625

Description: If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2353 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-alsb	⊢ (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Proof of Theorem bj-alsb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ala1 1741	. . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	a1d 25	. . 3 ⊢ (∀𝑥𝜑 → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	alrimiv 1855	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		df-ssb 32620	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	3, 4	sylibr 224	1 ⊢ (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4  ∀wal 1481  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-ssb 32620

This theorem is referenced by:  bj-ssbft  32642