Theorem sbft 2025

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

Assertion

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

Proof of Theorem sbft

Step	Hyp	Ref	Expression
1		sb1 1651	. . 3 ⊢ ([y / x]φ → ∃x(x = y ∧ φ))
2		simpr 447	. . . . 5 ⊢ ((x = y ∧ φ) → φ)
3	2	ax-gen 1546	. . . 4 ⊢ ∀x((x = y ∧ φ) → φ)
4		19.23t 1800	. . . 4 ⊢ (Ⅎxφ → (∀x((x = y ∧ φ) → φ) ↔ (∃x(x = y ∧ φ) → φ)))
5	3, 4	mpbii 202	. . 3 ⊢ (Ⅎxφ → (∃x(x = y ∧ φ) → φ))
6	1, 5	syl5 28	. 2 ⊢ (Ⅎxφ → ([y / x]φ → φ))
7		nfr 1761	. . 3 ⊢ (Ⅎxφ → (φ → ∀xφ))
8		stdpc4 2024	. . 3 ⊢ (∀xφ → [y / x]φ)
9	7, 8	syl6 29	. 2 ⊢ (Ⅎxφ → (φ → [y / x]φ))
10	6, 9	impbid 183	1 ⊢ (Ⅎxφ → ([y / x]φ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  [wsb 1648

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbf  2026  sbctt  3109