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Theorem hbsb2e 2518
Description: Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
hbsb2e ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Proof of Theorem hbsb2e
StepHypRef Expression
1 sb4e 2517 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2 sb2 2497 . . 3 (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → [𝑦 / 𝑥]∃𝑦𝜑)
32axc4i 2332 . 2 (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
41, 3syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526  wex 1771  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by: (None)
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