Theorem bj-ax12 34524

Description: Remove a DV condition from bj-ax12v 34523 (using core axioms only). (Contributed by BJ, 26-Dec-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax12	⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ax12

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ax12v 34523	. . 3 ⊢ ∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equequ2 2036	. . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
3	2	imbi1d 345	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
4	3	albidv 1928	. . . . . 6 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5	4	imbi2d 344	. . . . 5 ⊢ (𝑦 = 𝑡 → ((𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
6	2, 5	imbi12d 348	. . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))))
7	6	albidv 1928	. . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))))
8	1, 7	mpbii 236	. 2 ⊢ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
9		ax6ev 1978	. 2 ⊢ ∃𝑦 𝑦 = 𝑡
10	8, 9	exlimiiv 1939	1 ⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788

This theorem is referenced by:  bj-ax12ssb  34525  bj-subst  34528