Theorem sbcimi 38644

Description: Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)

Hypotheses

Ref	Expression
sbcimi.1	⊢ 𝐴 ∈ V
sbcimi.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
sbcimi.3	⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)

Assertion

Ref	Expression
sbcimi	⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Proof of Theorem sbcimi

Step	Hyp	Ref	Expression
1		sbcimi.1	. . 3 ⊢ 𝐴 ∈ V
2		sbcimg 3801	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
3	1, 2	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
4		sbcimi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒)
5		sbcimi.3	. . 3 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂)
6	4, 5	imbi12i 353	. 2 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) ↔ (𝜒 → 𝜂))
7	3, 6	bitri 278	1 ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2149  Vcvv 3463  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3754

This theorem is referenced by: (None)