Theorem 19.21v 1922

Description: Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as (𝜑 → ∀𝑥𝜑) in 19.21 1632 via the use of distinct variable conditions combined with ax-17 1575. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2085 derived from df-eu 2083. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
19.21v	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.21v

Step	Hyp	Ref	Expression
1		ax-17 1575	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	19.21h 1606	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm11.53  1945  cbval2  1971  cbvaldvaw  1980  sbhb  1994  2sb6  2038  sbcom2v  2039  2sb6rf  2044  2exsb  2063  moanim  2155  r3al  2586  ceqsralt  2841  rspc2gv  2933  euind  3004  reu2  3005  reuind  3022  unissb  3944  dfiin2g  4024  tfi  4704  asymref  5148  dff13  5941  mpo2eqb  6163