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Theorem axreplem 5159
Description: Lemma for axrep2 5160 and axrep3 5161. (Contributed by BJ, 6-Aug-2022.)
Assertion
Ref Expression
axreplem (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣   𝑥,𝑤   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜓(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem axreplem
StepHypRef Expression
1 elequ2 2127 . . . . . . 7 (𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))
21anbi1d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝑧𝑥𝜒) ↔ (𝑧𝑦𝜒)))
32exbidv 1922 . . . . 5 (𝑥 = 𝑦 → (∃𝑤(𝑧𝑥𝜒) ↔ ∃𝑤(𝑧𝑦𝜒)))
43bibi2d 346 . . . 4 (𝑥 = 𝑦 → ((𝜓 ↔ ∃𝑤(𝑧𝑥𝜒)) ↔ (𝜓 ↔ ∃𝑤(𝑧𝑦𝜒))))
54albidv 1921 . . 3 (𝑥 = 𝑦 → (∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒)) ↔ ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒))))
65imbi2d 344 . 2 (𝑥 = 𝑦 → ((𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ (𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))
76exbidv 1922 1 (𝑥 = 𝑦 → (∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑥𝜒))) ↔ ∃𝑢(𝜑 → ∀𝑣(𝜓 ↔ ∃𝑤(𝑧𝑦𝜒)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  axrep2  5160  axrep3  5161
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