Theorem exlimddvfi 33097

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 2413	. . 3 ⊢ (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
4	1, 3	sylib 207	. 2 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5		exlimddvfi.3	. 2 ⊢ Ⅎ𝑦𝜓
6		sbsbc 3406	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ [𝑦 / 𝑥]𝜃)
7		exlimddvfi.4	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
8	6, 7	bitri 263	. . 3 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
9		exlimddvfi.5	. . 3 ⊢ ((𝜂 ∧ 𝜓) → 𝜒)
10	8, 9	sylanb 488	. 2 ⊢ (([𝑦 / 𝑥]𝜃 ∧ 𝜓) → 𝜒)
11		exlimddvfi.6	. 2 ⊢ Ⅎ𝑦𝜒
12	4, 5, 10, 11	exlimddvf 33096	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699  [wsb 1867  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403

This theorem is referenced by: (None)