Theorem sbim 1982

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 1918	. . . 4 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
2	1	sbbii 1789	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓))
3		sbimv 1918	. . 3 ⊢ ([𝑦 / 𝑧]([𝑧 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
4	2, 3	bitri 184	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
5		ax-17 1550	. . 3 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓))
6	5	sbco2vh 1974	. 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
7		ax-17 1550	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
8	7	sbco2vh 1974	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
9		ax-17 1550	. . . 4 ⊢ (𝜓 → ∀𝑧𝜓)
10	9	sbco2vh 1974	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
11	8, 10	imbi12i 239	. 2 ⊢ (([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
12	4, 6, 11	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 105  [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbrim  1985  sblim  1986  sbbi  1988  moimv  2121  nfraldya  2542  sbcimg  3042  zfregfr  4627  tfi  4635  peano2  4648