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Theorem ax11v 1724
 Description: This is a version of ax-11o 1720 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 = 𝑦𝜑)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax11v
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 a9e 1602 . 2 𝑧 𝑧 = 𝑦
2 ax-17 1435 . . . . 5 (𝜑 → ∀𝑧𝜑)
3 ax-11 1413 . . . . 5 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
42, 3syl5 32 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
5 equequ2 1615 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
65imbi1d 224 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
76albidv 1721 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
87imbi2d 223 . . . . 5 (𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8imbi12d 227 . . . 4 (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
104, 9mpbii 140 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1110exlimiv 1505 . 2 (∃𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
121, 11ax-mp 7 1 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257   = wceq 1259  ∃wex 1397 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-17 1435  ax-i9 1439 This theorem depends on definitions:  df-bi 114 This theorem is referenced by:  equs5or  1727  sb56  1781
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