Theorem sbcim2g 44512

Description: Distribution of class substitution over a left-nested implication. Similar to sbcimg 3793. sbcim2g 44512 is sbcim2gVD 44848 without virtual deductions and was automatically derived from sbcim2gVD 44848 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sbcim2g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Proof of Theorem sbcim2g

Step	Hyp	Ref	Expression
1		sbcimg 3793	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
2	1	biimpd 229	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
3		sbcimg 3793	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)))
4		imbi2 348	. . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
5	4	biimpcd 249	. . 3 ⊢ (([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒)) → (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
6	2, 3, 5	syl6ci 71	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
7		idd 24	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))
8		biimpr 220	. . . 4 ⊢ (([𝐴 / 𝑥](𝜓 → 𝜒) ↔ ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → (([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒) → [𝐴 / 𝑥](𝜓 → 𝜒)))
9	3, 7, 8	ee13 44478	. . 3 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))))
10	9, 1	sylibrd 259	. 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))))
11	6, 10	impbid 212	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sbc 3745

This theorem is referenced by:  trsbc  44514  trsbcVD  44850