Theorem sbctt 3109

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
sbctt	⊢ ((A ∈ V ∧ Ⅎxφ) → ([̣A / x]̣φ ↔ φ))

Proof of Theorem sbctt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. . . . 5 ⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ))
2	1	bibi1d 310	. . . 4 ⊢ (y = A → (([y / x]φ ↔ φ) ↔ ([̣A / x]̣φ ↔ φ)))
3	2	imbi2d 307	. . 3 ⊢ (y = A → ((Ⅎxφ → ([y / x]φ ↔ φ)) ↔ (Ⅎxφ → ([̣A / x]̣φ ↔ φ))))
4		sbft 2025	. . 3 ⊢ (Ⅎxφ → ([y / x]φ ↔ φ))
5	3, 4	vtoclg 2915	. 2 ⊢ (A ∈ V → (Ⅎxφ → ([̣A / x]̣φ ↔ φ)))
6	5	imp 418	1 ⊢ ((A ∈ V ∧ Ⅎxφ) → ([̣A / x]̣φ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3047

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by:  sbcgf  3110  csbtt  3149