Theorem r19.29d2r 3228

Description: Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
r19.29d2r.1	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
r19.29d2r.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)

Assertion

Ref	Expression
r19.29d2r	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Proof of Theorem r19.29d2r

Step	Hyp	Ref	Expression
1		r19.29d2r.1	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
2		r19.29d2r.2	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
3		r19.29 3220	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
4	1, 2, 3	syl2anc 567	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒))
5		r19.29 3220	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
6	5	reximi 3159	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑦 ∈ 𝐵 𝜒) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
7	4, 6	syl 17	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 382  ∀wral 3061  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ral 3066  df-rex 3067

This theorem is referenced by:  r19.29vva  3229  ucnima  22306  tgisline  25744  r19.29ffa  29661  rnmpt2ss  29814  xrofsup  29874  icoreresf  33538