Theorem 19.21 1594

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.21.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem 19.21

Step	Hyp	Ref	Expression
1		19.21.1	. 2 ⊢ Ⅎ𝑥𝜑
2		19.21t 1593	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-4 1521  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  stdpc5  1595  19.21-2  1678  19.32dc  1690  cbv1  1756  cbv1v  1758  eu2  2082  mo3h  2091  moanim  2112  r2alf  2507