Theorem sbc19.21g 3535

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Hypothesis

Ref	Expression
sbcgf.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
sbc19.21g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))

Proof of Theorem sbc19.21g

Step	Hyp	Ref	Expression
1		sbcimg 3510	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
2		sbcgf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	sbcgf 3534	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	3	imbi1d 330	. 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))
5	1, 4	bitrd 268	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1748   ∈ wcel 2030  [wsbc 3468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-v 3233  df-sbc 3469

This theorem is referenced by:  bnj121  31066  bnj124  31067  bnj130  31070  bnj207  31077  bnj611  31114  bnj1000  31137