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Theorem r19.45zv 3647
 Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45zv (A → (x A (φ ψ) ↔ (φ x A ψ)))
Distinct variable groups:   x,A   φ,x
Allowed substitution hint:   ψ(x)

Proof of Theorem r19.45zv
StepHypRef Expression
1 r19.9rzv 3644 . . 3 (A → (φx A φ))
21orbi1d 683 . 2 (A → ((φ x A ψ) ↔ (x A φ x A ψ)))
3 r19.43 2766 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
42, 3syl6rbbr 255 1 (A → (x A (φ ψ) ↔ (φ x A ψ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ≠ wne 2516  ∃wrex 2615  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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