Theorem reximddv 2636

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

Ref	Expression
reximddva.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
reximddva.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
reximddv	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem reximddv

Step	Hyp	Ref	Expression
1		reximddva.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		reximddva.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
3	2	expr 375	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
4	3	reximdva 2635	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
5	1, 4	mpd 13	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∃wrex 2512

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-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  reximddv2  2638