Theorem sbim 1946

Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)

Assertion

Ref	Expression
sbim	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbim

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbimv 1886	. . . 4 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
2	1	sbbii 1758	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
3		sbimv 1886	. . 3 ⊢ ([𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
4	2, 3	bitri 183	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
5		ax-17 1519	. . 3 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓))
6	5	sbco2vh 1938	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
7		ax-17 1519	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
8	7	sbco2vh 1938	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9		ax-17 1519	. . . 4 ⊢ (𝜓 → ∀𝑧𝜓)
10	9	sbco2vh 1938	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
11	8, 10	imbi12i 238	. 2 ⊢ (([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
12	4, 6, 11	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 104  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbrim  1949  sblim  1950  sbbi  1952  moimv  2085  nfraldya  2505  sbcimg  2996  zfregfr  4558  tfi  4566  peano2  4579