Theorem bj-ax12ssb 37321

Description: Axiom bj-ax12 37320 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax12ssb	⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥, 𝑡)

Proof of Theorem bj-ax12ssb

Step	Hyp	Ref	Expression
1		bj-ax12 37320	. . 3 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2		sb6 2122	. . . . . 6 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
3	2	imbi2i 339	. . . . 5 ⊢ ((𝜑 → [𝑡 / 𝑥]𝜑) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
4	3	imbi2i 339	. . . 4 ⊢ ((𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
5	4	albii 1852	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
6	1, 5	mpbir 234	. 2 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))
7		sb6 2122	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → [𝑡 / 𝑥]𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)))
8	6, 7	mpbir 234	1 ⊢ [𝑡 / 𝑥](𝜑 → [𝑡 / 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)