Theorem rexxfr 5086

Description: Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)

Hypotheses

Ref	Expression
ralxfr.1	⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
ralxfr.2	⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
ralxfr.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexxfr	⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓)

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem rexxfr

Step	Hyp	Ref	Expression
1		dfrex2 3176	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜑)
2		dfrex2 3176	. . 3 ⊢ (∃𝑦 ∈ 𝐶 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐶 ¬ 𝜓)
3		ralxfr.1	. . . 4 ⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)
4		ralxfr.2	. . . 4 ⊢ (𝑥 ∈ 𝐵 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴)
5		ralxfr.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	notbid 310	. . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓))
7	3, 4, 6	ralxfr 5084	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐶 ¬ 𝜓)
8	2, 7	xchbinxr 327	. 2 ⊢ (∃𝑦 ∈ 𝐶 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜑)
9	1, 8	bitr4i 270	1 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐶 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  ∀wral 3089  ∃wrex 3090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387

This theorem is referenced by:  infm3  11274  reeff1o  24542  moxfr  38037