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Theorem drnf2 2468
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)
Hypothesis
Ref Expression
dral1.1 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
drnf2 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Proof of Theorem drnf2
StepHypRef Expression
1 nfae 2456 . 2 𝑧𝑥 𝑥 = 𝑦
2 dral1.1 . 2 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))
31, 2nfbidf 2237 1 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857
This theorem is referenced by:  nfsb4t  2524  drnfc2  2919
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