Theorem sbctt 3806

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 3739	. . . . 5 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	bibi1d 343	. . . 4 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ 𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)))
3	2	imbi2d 340	. . 3 ⊢ (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))))
4		sbft 2272	. . 3 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
5	3, 4	vtoclg 3507	. 2 ⊢ (𝐴 ∈ 𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)))
6	5	imp 406	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  Ⅎwnf 1784  [wsb 2067   ∈ wcel 2111  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by:  sbcgf  3807  csbtt  3862