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Theorem ax11v 1708
 Description: This is a version of ax-11o 1704 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 1586 . 2 𝑧 𝑧 = 𝑦
2 ax-17 1419 . . . . 5 (𝜑 → ∀𝑧𝜑)
3 ax-11 1397 . . . . 5 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
42, 3syl5 28 . . . 4 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
5 equequ2 1599 . . . . 5 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
65imbi1d 220 . . . . . . 7 (𝑧 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑥 = 𝑦𝜑)))
76albidv 1705 . . . . . 6 (𝑧 = 𝑦 → (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
87imbi2d 219 . . . . 5 (𝑧 = 𝑦 → ((𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
95, 8imbi12d 223 . . . 4 (𝑧 = 𝑦 → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))))
104, 9mpbii 136 . . 3 (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
1110exlimiv 1489 . 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 1241   = wceq 1243  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-17 1419  ax-i9 1423 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  equs5or  1711  sb56  1765
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