Theorem sbc19.21g 3814

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 3791	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
2		sbcgf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	sbcgf 3813	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
4	3	imbi1d 341	. 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))
5	1, 4	bitrd 279	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206  Ⅎwnf 1785   ∈ wcel 2114  [wsbc 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3743

This theorem is referenced by:  bnj121  35046  bnj124  35047  bnj130  35050  bnj207  35057  bnj611  35094  bnj1000  35117