Theorem r19.26-2 2663

Description: Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)

Assertion

Ref	Expression
r19.26-2	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Proof of Theorem r19.26-2

Step	Hyp	Ref	Expression
1		r19.26 2660	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
2	1	ralbii 2539	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓))
3		r19.26 2660	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
4	2, 3	bitri 184	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∀wral 2511

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-4 1559  ax-17 1575

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-ral 2516

This theorem is referenced by:  fununi  5405  issgrpv  13550  issgrpn0  13551  isnsg2  13853  dfrhm2  14232  df2idl2rng  14587