Theorem sbft 2265

Description: Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)

Assertion

Ref	Expression
sbft	⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))

Proof of Theorem sbft

Step	Hyp	Ref	Expression
1		spsbe 2086	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2		19.9t 2200	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	syl5ib 243	. 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝜑))
4		nf5r 2189	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		stdpc4 2072	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
6	4, 5	syl6 35	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
7	3, 6	impbid 211	1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788  df-sb 2069

This theorem is referenced by:  sbf  2266  sbctt  3788  wl-sbrimt  35632  wl-sblimt  35633  wl-equsb4  35639  ichnfimlem  44803