Theorem bj-ax12ssb 31824

Description: The axiom bj-ax12 31823 expressed using substitution. (Contributed by BJ, 26-Dec-2020.)

Assertion

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

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12ssb

Step	Hyp	Ref	Expression
1		bj-ax12 31823	. . 3 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
2		bj-ssb1 31822	. . . . . 6 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
3	2	imbi2i 325	. . . . 5 ⊢ ((𝜑 → [𝑡/𝑥]b𝜑) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
4	3	imbi2i 325	. . . 4 ⊢ ((𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
5	4	albii 1737	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
6	1, 5	mpbir 220	. 2 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑))
7		bj-ssb1 31822	. 2 ⊢ ([𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)))
8	6, 7	mpbir 220	1 ⊢ [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4  ∀wal 1473  [wssb 31808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ssb 31809

This theorem is referenced by: (None)