Theorem rexbiOLD 3174

Description: Obsolete version of rexbi 3173 as of 31-Oct-2024. (Contributed by Scott Fenton, 7-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rexbiOLD	⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem rexbiOLD

Step	Hyp	Ref	Expression
1		ralbi 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 ↔ ¬ 𝜓) → (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
2	1	notbid 318	. 2 ⊢ (∀𝑥 ∈ 𝐴 (¬ 𝜑 ↔ ¬ 𝜓) → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
3		notbi 319	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓))
4	3	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (¬ 𝜑 ↔ ¬ 𝜓))
5		dfrex2 3170	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
6		dfrex2 3170	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓)
7	5, 6	bibi12i 340	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓) ↔ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜓))
8	2, 4, 7	3imtr4i 292	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀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: (None)