Theorem sbcimi 38779

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 3792	. . 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 2143  Vcvv 3455  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sbc 3745

This theorem is referenced by: (None)