Theorem ax11ev 1815

Description: Analogue to ax11v 1814 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)

Assertion

Ref	Expression
ax11ev	⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11ev

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1683	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		ax11e 1783	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) → ∃𝑧𝜑))
3		ax-17 1513	. . . . . 6 ⊢ (𝜑 → ∀𝑧𝜑)
4	3	19.9h 1630	. . . . 5 ⊢ (∃𝑧𝜑 ↔ 𝜑)
5	2, 4	syl6ib 160	. . . 4 ⊢ (𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) → 𝜑))
6		equequ2 1700	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
7	6	anbi1d 461	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜑)))
8	7	exbidv 1812	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
9	8	imbi1d 230	. . . . 5 ⊢ (𝑧 = 𝑦 → ((∃𝑥(𝑥 = 𝑧 ∧ 𝜑) → 𝜑) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑)))
10	6, 9	imbi12d 233	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (∃𝑥(𝑥 = 𝑧 ∧ 𝜑) → 𝜑)) ↔ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))))
11	5, 10	mpbii 147	. . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑)))
12	11	exlimiv 1585	. 2 ⊢ (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑)))
13	1, 12	ax-mp 5	1 ⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1342  ∃wex 1479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)