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Theorem sbieh 1715
Description: Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1716 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.)
Hypotheses
Ref Expression
sbieh.1 (𝜓 → ∀𝑥𝜓)
sbieh.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
sbieh ([𝑦 / 𝑥]𝜑𝜓)

Proof of Theorem sbieh
StepHypRef Expression
1 id 19 . 2 (𝜑𝜑)
21hbth 1393 . . 3 ((𝜑𝜑) → ∀𝑥(𝜑𝜑))
3 sbieh.1 . . . 4 (𝜓 → ∀𝑥𝜓)
43a1i 9 . . 3 ((𝜑𝜑) → (𝜓 → ∀𝑥𝜓))
5 sbieh.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
65a1i 9 . . 3 ((𝜑𝜑) → (𝑥 = 𝑦 → (𝜑𝜓)))
72, 4, 6sbiedh 1712 . 2 ((𝜑𝜑) → ([𝑦 / 𝑥]𝜑𝜓))
81, 7ax-mp 7 1 ([𝑦 / 𝑥]𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1283  [wsb 1687
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-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1688
This theorem is referenced by:  sbie  1716  sbco2vlem  1863  equsb3lem  1867  sbco2yz  1880  elsb3  1895  elsb4  1896  dvelimf  1934
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