Theorem ax11v 1827

Description: This is a version of ax-11o 1823 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)

Assertion

Ref	Expression
ax11v	⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11v

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1696	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		ax-17 1526	. . . . 5 ⊢ (𝜑 → ∀𝑧𝜑)
3		ax-11 1506	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
4	2, 3	syl5 32	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
5		equequ2 1713	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
6	5	imbi1d 231	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
7	6	albidv 1824	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8	7	imbi2d 230	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
9	5, 8	imbi12d 234	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
10	4, 9	mpbii 148	. . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
11	10	exlimiv 1598	. 2 ⊢ (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
12	1, 11	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4  ∀wal 1351   = wceq 1353  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-17 1526  ax-i9 1530

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equs5or  1830  sb56  1885