Theorem exlimddvfi 33927

Description: A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)

Hypotheses

Ref	Expression
exlimddvfi.1	⊢ (𝜑 → ∃𝑥𝜃)
exlimddvfi.2	⊢ Ⅎ𝑦𝜃
exlimddvfi.3	⊢ Ⅎ𝑦𝜓
exlimddvfi.4	⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
exlimddvfi.5	⊢ ((𝜂 ∧ 𝜓) → 𝜒)
exlimddvfi.6	⊢ Ⅎ𝑦𝜒

Assertion

Ref	Expression
exlimddvfi	⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Proof of Theorem exlimddvfi

Step	Hyp	Ref	Expression
1		exlimddvfi.1	. . 3 ⊢ (𝜑 → ∃𝑥𝜃)
2		exlimddvfi.2	. . . 4 ⊢ Ⅎ𝑦𝜃
3	2	sb8e 2425	. . 3 ⊢ (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
4	1, 3	sylib 208	. 2 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5		exlimddvfi.3	. 2 ⊢ Ⅎ𝑦𝜓
6		sbsbc 3439	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ [𝑦 / 𝑥]𝜃)
7		exlimddvfi.4	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
8	6, 7	bitri 264	. . 3 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
9		exlimddvfi.5	. . 3 ⊢ ((𝜂 ∧ 𝜓) → 𝜒)
10	8, 9	sylanb 489	. 2 ⊢ (([𝑦 / 𝑥]𝜃 ∧ 𝜓) → 𝜒)
11		exlimddvfi.6	. 2 ⊢ Ⅎ𝑦𝜒
12	4, 5, 10, 11	exlimddvf 33926	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1704  Ⅎwnf 1708  [wsb 1880  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by: (None)