Theorem r19.29imd 3186

Description: Theorem 19.29 of [Margaris] p. 90 with an implication in the hypothesis containing the generalization, deduction version. (Contributed by AV, 19-Jan-2019.)

Hypotheses

Ref	Expression
r19.29imd.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
r19.29imd.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))

Assertion

Ref	Expression
r19.29imd	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Proof of Theorem r19.29imd

Step	Hyp	Ref	Expression
1		r19.29imd.1	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		r19.29imd.2	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
3		r19.29r 3185	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)))
5		abai 824	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ (𝜓 → 𝜒)))
6	5	rexbii 3181	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)))
7	4, 6	sylibr 233	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 396  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069  df-rex 3070

This theorem is referenced by:  psgndif  20807  neik0pk1imk0  41657