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Theorem equs45f 1725
Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.)
Hypothesis
Ref Expression
equs45f.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
equs45f (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))

Proof of Theorem equs45f
StepHypRef Expression
1 equs45f.1 . . . . 5 (𝜑 → ∀𝑦𝜑)
21anim2i 334 . . . 4 ((𝑥 = 𝑦𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
32eximi 1532 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
4 equs5a 1717 . . 3 (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
53, 4syl 14 . 2 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
6 equs4 1655 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))
75, 6impbii 124 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-11 1438  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  sb5f  1727
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