Theorem exlimddvfi 39246

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 2550	. . 3 ⊢ (∃𝑥𝜃 ↔ ∃𝑦[𝑦 / 𝑥]𝜃)
4	1, 3	sylib 221	. 2 ⊢ (𝜑 → ∃𝑦[𝑦 / 𝑥]𝜃)
5		exlimddvfi.3	. 2 ⊢ Ⅎ𝑦𝜓
6		sbsbc 3746	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ [𝑦 / 𝑥]𝜃)
7		exlimddvfi.4	. . . 4 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
8	6, 7	bitri 278	. . 3 ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂)
9		exlimddvfi.5	. . 3 ⊢ ((𝜂 ∧ 𝜓) → 𝜒)
10	8, 9	sylanb 593	. 2 ⊢ (([𝑦 / 𝑥]𝜃 ∧ 𝜓) → 𝜒)
11		exlimddvfi.6	. 2 ⊢ Ⅎ𝑦𝜒
12	4, 5, 10, 11	exlimddvf 39245	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812  Ⅎwnf 1816  [wsb 2101  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by: (None)