Theorem rmoeqd 2637

Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
raleqd.1	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rmoeqd	⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoeqd

Step	Hyp	Ref	Expression
1		rmoeq1 2629	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑))
2		raleqd.1	. . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	rmobidv 2619	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐵 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
4	1, 3	bitrd 187	1 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ∃*wrmo 2419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rmo 2424

This theorem is referenced by: (None)